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TREATISE mmm
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xSm*^i?c?^n
A ON THE DIFFERENTIAL GEOMETRY OF PURVES AND SURFACES
BY
LUTHER PFAHLER EISENHART
PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY
GINN AND COMPANY
BOSTON
NEW YORK CHICAGO LONDON
CvA
COPYRIGHT,
1909,
BY
LUTHER PFAHLER EJSENHART
ALL RIGHTS RESERVED
898
gftc
SUftengum
PKOU.S.A.
GINN AND COMPANY PRILTORS BOSTON
6
A.ATH..
STAT.
LIBRARY
PEEFACE
This book
feel that
is
a development from courses which I have given in
number of years. During this time I have come to more would be accomplished by my students if they had an otherwise adapted to the introductory treatise written in English and
Princeton for a
use of
men
beginning their graduate work.
the method Chapter I is devoted to the theory of twisted curves, in general being that which is usually followed in discussions of this I have introduced the idea of moving axes, subject. But in addition and have derived the formulas pertaining thereto from the previously
In this way the student is made familiar with a method which is similar to that used by Darboux in the first volume of his Lemons, and to that of Cesaro in his Geometria
obtained FrenetSerret formulas.
not only of great advantage in the treat ment of certain topics and in the solution of problems, but it is valu
Intrinseca.
This method
is
able in developing geometrical thinking.
book may be divided into three parts. The deals with the geometry of a sur first, consisting of Chapters IIYI, face in the neighborhood of a point and the developments therefrom,
The remainder
of the
such as curves and systems
tions.
of curves defined
by
differential
equa
by large extent the method from the discussion of two quad properties of a surface are derived ratic differential forms. However, little or no space is given to the
To a
is that of Gauss,
which the
and their invariants. In algebraic treatment of differential forms as defined in the first chapter, addition, the method of moving axes, has been extended so as to be applicable to an investigation of the
properties
of
surfaces
and groups
of
is
surfaces.
The extent
of the
no attempt has theory concerning ordinary points consider the exceptional problems. For a discussion been made to of such questions as the existence of integrals of differential equa tions and boundary conditions the reader must consult the treatises which deal particularly with these subjects. In Chapters VII and VIII the theory previously developed is as the quadrics, ruled applied to several groups of surfaces, such minimal surfaces, surfaces of constant total curvature, and surfaces,
so great that
surfaces with plane
and spherical
lines of curvature.
iii
iv
PEEFACE
The idea
of applicability of surfaces is introduced in Chapter III as a particular case of conformal representation, and throughout the
book attention
called to examples of applicable surfaces. However, the general problems concerned with the applicability of surfaces are discussed in Chapters IX and X, the latter of which deals entirely
is
with the recent method
of
Weingarten and
its
developments.
The
remaining four chapters are devoted to a discussion of infinitesimal
deformation of surfaces, congruences of straight lines and of
circles,
and
triply orthogonal systems of surfaces. It will be noticed that the book contains
many
them
examples, and the
are merely direct
student will find that
applications of
w hereas
r
certain of
formulas, theory which might properly be included as portions of a more ex tensive treatise. At first I felt constrained to give such references as
the
others
constitute
extensions of the
would enable the reader to consult the journals and treatises from which some of these problems were taken, but finally it seemed best to furnish no such key, only to remark that the Encyklopadie der mathematisclicn Wissenschaften may be of assistance. And the same
may
the book.
be said about references to the sources of the subjectmatter of Many important citations have been made, but there has
not been an attempt to give every reference. However, I desire to acknowledge my indebtedness to the treatises of Darboux, Bianchi, and Scheffers. But the difficulty is that for many years I have con
sulted these authors so freely that now it is impossible for except in certain cases, what specific debts I owe to each.
me
to say,
In its present form, the material of the first eight chapters has been given to beginning classes in each of the last two years; and the remainder of the book, with certain enlargements, has constituted an advanced course which has been followed several times. It is im suitable credit for the suggestions made and for me to
give possible the assistance rendered by my students during these years, but I am conscious of helpful suggestions made by my colleagues, Professors Veblen, Maclnnes, and Swift, and by my former colleague, Professor
Bliss of Chicago. I wish also to thank Mr. A. K. Krause for making the drawings for the figures. It remains for me to express my appreciation of the courtesy shown by Ginn and Company, and of the assistance given by them during the printing of this book.
LUTHER PFAHLER E1SENHART
CONTENTS
CHAPTEE
SECTION
1.
I
CURVES IN SPACE
PAGE
1
2. 3. 4.
5. 6.
7.
8.
9.
10.
11.
12.
PARAMETRIC EQUATIONS OF A CURVE OTHER FORMS OF THE EQUATIONS OF A CURVE LINEAR ELEMENT TANGENT TO A CURVE ORDER OF CONTACT. NORMAL PLANK CURVATURE. RADIUS OF FIRST CURVATURE OSCULATING PLANE PRINCIPAL NORMAL AND BINORMAL OSCULATING CIRCLE. CENTER OF FIRST CURVATURE TORSION. FRENETSERRET FORMULAS FORM OF CURVE IN THE NEIGHBORHOOD OF A POINT. THE SIGN OF TORSION CYLINDRICAL HELICES
3
4
6
8
9
10 12
14
....
16
18
20
22
13.
INTRINSIC EQUATIONS.
FUNDAMENTAL THEOREM
14. 15.
RICCATI EQUATIONS
25
THE DETERMINATION OF THE COORDINATES OF A CURVE DEFINED
BY
ITS INTRINSIC
EQUATIONS
27 30
16. 17.
MOVING TRIHEDRAL ILLUSTRATIVE EXAMPLES
OSCULATING SPHERE
.33
37
39 41
18.
19.
20.
21. 22.
BERTRAND CURVES TANGENT SURFACE OF A CURVE INVOLUTES AND EVOLUTES OF A CURVE MINIMAL CURVES
43
.
47
CHAPTER
II
CURVILINEAR COORDINATES ON A SURFACE. ENVELOPES
23.
24.
25. 26.
PARAMETRIC EQUATIONS OF A SURFACE PARAMETRIC CURVES TANGENT PLANE ONEPARAMETER FAMILIES OF SURFACES. ENVELOPES
v
52 54
56
....
59
vi
SECTION
27.
28.
29.
CONTENTS
PAGE
DEVELOPABLE SURFACES. RECTIFYING DEVELOPABLE APPLICATIONS OF THE MOVING TRIHEDRAL ENVELOPE OF SPHERES. CANAL SURFACES
....
61 04
66
CHAPTER
III
LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAME TERS. CONFORMAL REPRESENTATION
30.
LINEAR ELEMENT
ISOTROPIC DEVELOPABLE
70
72
72
31. 32. 33. 34.
35.
36.
37. 38. 39.
40. 41.
42. 43. 44.
45.
46. 47.
TRANSFORMATION OF COORDINATES ANGLES BETWEEN CURVES. THE ELEMENT OF AREA FAMILIES OF CURVES MINIMAL CURVES ON A SURFACE VARIATION OF A FUNCTION DIFFERENTIAL PARAMETERS OF THE FIRST ORDER DIFFERENTIAL PARAMETERS OF THE SECOND ORDER SYMMETRIC COORDINATES ISOTHERMIC AND ISOMETRIC PARAMETERS ISOTHERMIC ORTHOGONAL SYSTEMS CONFORMAL REPRESENTATION ISOMETRIC REPRESENTATION. APPLICABLE SURFACES CONFORMAL REPRESENTATION OF A SURFACE UPON ITSELF CONFORMAL REPRESENTATION OF THE PLANE SURFACES OF REVOLUTION CONFORMAL REPRESENTATIONS OF THE SPHERE
.
....
74
78
81
82
84
....
88
91
93
95
98
....
.
.
100
101
104 107
109
CHAPTER IV
GEOMETRY OF A SURFACE IN THE NEIGHBORHOOD
OF A POINT
48.
49.
50.
51. 52. 53. 54.
55.
56.
FUNDAMENTAL COEFFICIENTS OF THE SECOND ORDER RADIUS OF NORMAL CURVATURE PRINCIPAL RADII OF NORMAL CURVATURE
LINES OF CURVATURE.
....
114
117
118
121
EQUATIONS OF RODRIGUES
TOTAL AND MEAN CURVATURE EQUATION OF EULER. DUPIN INDICATRIX CONJUGATE DIRECTIONS AT A POINT. CONJUGATE SYSTEMS
ASYMPTOTIC LINES. CHARACTERISTIC LINES CORRESPONDING SYSTEMS ON Two SURFACES GEODESIC CURVATURE. GEODESICS
123
124
.
126 128
.......
...
130
131 133
57.
58.
FUNDAMENTAL FORMULAS
.
CONTENTS
SECTION
59.
GO.
61.
vii
PAGE
137
141
62.
GEODESIC TORSION SPHERICAL REPRESENTATION RELATIONS BETWEEN A SURFACE AND ITS SPHERICAL REPRE SENTATION HELICOIDS
143 146
CHAPTEE V
FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL
63.
ClIRISTOFFEL SYMBOLS
152
64.
65. 66. 67.
68.
THE EQUATIONS OF GAUSS AND OF CODAZZI FUNDAMENTAL THEOREM FUNDAMENTAL EQUATIONS IN ANOTHER FORM
TANGENTIAL COORDINATES. MEAN EVOLUTE THE MOVING TRIHEDRAL FUNDAMENTAL EQUATIONS OF CONDITION LINEAR ELEMENT. LINES OF CURVATURE CONJUGATE DIRECTIONS AND ASYMPTOTIC DIRECTONS. SPHER ICAL REPRESENTATION FUNDAMENTAL RELATIONS AND FORMULAS PARALLEL SURFACES SURFACES OF CENTER FUNDAMENTAL QUANTITIES FOR SURFACES OF CENTER SURFACES COMPLEMENTARY TO A GIVEN SURFACE
. .
153
157
160
162 166
69.
168
171 172
70. 71.
72.
73. 74. 75.
76.
174
177
179
.
181
184
CHAPTER VI
SYSTEMS OF CURVES. GEODESICS
77.
78. 79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89. 90.
ASYMPTOTIC LINES SPHERICAL REPRESENTATION OF ASYMPTOTIC LINES FORMULAS OF LELIEUVRE. TANGENTIAL EQUATIONS CONJUGATE SYSTEMS OF PARAMETRIC LINES. INVERSIONS SURFACES OF TRANSLATION ISOTHERMALCONJUGATE SYSTEMS SPHERICAL REPRESENTATION OF CONJUGATE SYSTEMS TANGENTIAL COORDINATES. PROJECTIVE TRANSFORMATIONS EQUATIONS OF GEODESIC LINES GEODESIC PARALLELS. GEODESIC PARAMETERS GEODESIC POLAR COORDINATES AREA OF A GEODESIC TRIANGLE LINES OF SHORTEST LENGTH. GEODESIC CURVATURE GEODESIC ELLIPSES AND HYPERBOLAS
.
189
191
.... ....
. .
193
195
197
198
. .
.
200 201 204
206
207
209
....
. .
212
213
viii
CONTENTS
PAGE
214
. . .
SECTION
91.
92.
93.
94.
SURFACES OF LIOUVILLE INTEGRATION OF THE EQUATION OF GEODESIC LINES GEODESICS ON SURFACES OF LIOUVILLE LINES OF SHORTEST LENGTH. ENVELOPE OF GEODESICS
215
218
220
CHAPTEK
VII
QUADRICS. RULED SURFACES. MINIMAL SURFACES
95.
96.
97.
98.
99.
100.
101. 102.
103. 104. 105. 106.
107.
108.
109.
110. 111.
112. 113.
114.
CONFOCAL QUADRICS. ELLIPTIC COORDINATES FUNDAMENTAL QUANTITIES FOR CENTRAL QUADRICS FUNDAMENTAL QUANTITIES FOR THE PARABOLOIDS LINES OF CURVATURE AND ASYMPTOTIC LINES ox QUADRICS GEODESICS ON QUADRICS GEODESICS THROUGH THE UMBILICAL POINTS ELLIPSOID REFERRED TO A POLAR GEODESIC SYSTEM PROPERTIES OF QUADRICS EQUATIONS OF A RULED SURFACE LINE OF STRICTION. DEVELOPABLE SURFACES CENTRAL PLANE. PARAMETER OF DISTRIBUTION PARTICULAR FORM OF THE LINEAR ELEMENT ASYMPTOTIC LINES. ORTHOGONAL PARAMETRIC SYSTEMS MINIMAL SURFACES LINES OF CURVATURE AND ASYMPTOTIC LINES. ADJOINT MINI MAL SURFACES MINIMAL CURVES ON A MINIMAL SURFACE DOUBLE MINIMAL SURFACES ALGEBRAIC MINIMAL SURFACES ASSOCIATE SURFACES FORMULAS OF SCHWARZ
.
. .
.
226
.
229
....
.
230
232 231
236
.
236
239
241
242
244 247
.
.
248
250
253
254
258
260
263 264
.
r
CHAPTER
VIII
SURFACES OF CONSTANT TOTAL CURVATURE. TFSURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CUR
VATURE
115.
116.
117.
118.
119.
120. 121.
SPHERICAL SURFACES OF REVOLUTION PSEUDOSPHERICAL SURFACES OF REVOLUTION GEODESIC PARAMETRIC SYSTEMS. APPLICABILITY TRANSFORMATION OF HAZZIDAKIS TRANSFORMATION OF BIANCHI TRANSFORMATION OF BACKLUND THEOREM OF PERMUTABILITY
270
272 275
278 280
284
286
CONTENTS
SECTION
122.
123.
ix
PAGE
.
.
TRANSFORMATION OF LIE
JFSURFACES.
289
291
FUNDAMENTAL QUANTITIES
.
124.
125.
126.
127.
128.
129.
EVOLUTE OF A TFSuitFACE SURFACES OF CONSTANT MEAN CURVATURE RULED IFSuRFACES SPHERICAL REPRESENTATION OF SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN BOTH SYSTEMS SURFACES WITH PLANE LINES OF CURVATURE IN ONE SYSTEM.
294
.......
296
299
300
302
SURFACES OF MONGE
130.
305
307
308
131.
132.
133.
134.
MOLDING SURFACES SURFACES OF JOACHIMSTHAL SURFACES WITH CIRCULAR LINES OF CURVATURE CYCLIDES OF DUPIN SURFACES WITH SPHERICAL LINES OF CURVATURE
SYSTEM
310
312
IN
ONE
314
CHAPTER IX
DEFORMATION OF SURFACES
135.
136. 137.
138.
139.
140.
141.
142.
PROBLEM OF MINDING. SURFACES OF CONSTANT CURVATURE SOLUTION OF THE PROBLEM OF MINDING DEFORMATION OF MINIMAL SURFACES SECOND GENERAL PROBLEM OF DEFORMATION DEFORMATIONS WHICH CHANGE A CURVE ON THE SURFACE INTO A GIVEN CURVE IN SPACE LINES OF CURVATURE IN CORRESPONDENCE CONJUGATE SYSTEMS IN CORRESPONDENCE ASYMPTOTIC LINES IN CORRESPONDENCE. DEFORMATION OF A
.
321
323
327
331
333
336 338
143.
RULED SURFACE METHOD OF MINDING
PARTICULAR DEFORMATIONS OF RULED SURFACES
342
344
144.
345
CHAPTER X
DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN
145.
146.
147.
148.
REDUCED FORM OF THE LINEAR ELEMENT GENERAL FORMULAS THE THEOREM OF WEINGARTEN OTHER FORMS OF THE THEOREM OF WEINGARTEN
351
....
.
149.
SURFACES APPLICABLE TO A SURFACE OF REVOLUTION
.
353 355 357 362
x
SECTION
150. 151.
CONTENTS
PAGE
364
MINIMAL LINES ON THE SPHERE PARAMETRIC SURFACES OF GOURSAT. SURFACES APPLICABLE TO CERTAIN PARABOLOIDS
366
CHAPTER XI
INFINITESIMAL DEFORMATION OF SURFACES
152. 153. 154.
GENERAL PROBLEM
CHARACTERISTIC FUNCTION ASYMPTOTIC LINES PARAMETRIC ASSOCIATE SURFACES
....
373 374
376
155.
156. 157. 158.
159.
378
379
PARTICULAR PARAMETRIC CURVES RELATIONS BETWEEN THREE SURFACES S, S v S SURFACES RESULTING FROM AN INFINITESIMAL DEFORMATION ISOTHERMIC SURFACES
382
385
387
CHAPTER
XII
RECTILINEAR CONGRUENCES
160.
161. 162.
DEFINITION OF A CONGRUENCE.
LIMIT POINTS.
SPHERICAL REPRESENTATION
392
393 395
398
NORMAL CONGRUENCES. RULED SURFACES OF A CONGRUENCE
FOCAL SURFACES
PRINCIPAL SURFACES 163. DEVELOPABLE SURFACES OF A CONGRUENCE. 164. ASSOCIATE NORMAL CONGRUENCES
165. 166.
167.
401
168. 169.
170. 171.
172.
173.
DERIVED CONGRUENCES FUNDAMENTAL EQUATIONS OF CONDITION SPHERICAL REPRESENTATION OF PRINCIPAL SURFACES AND OF DEVELOPABLES FUNDAMENTAL QUANTITIES FOR THE FOCAL SURFACES ISOTROPIC CONGRUENCES CONGRUENCES OF GUICIIARD PSEUDOSPHERICAL CONGRUENCES ITCONGRUENCES CONGRUENCES OF RIBAUCOUR
........ ......
. .
.
403
406
407 409 412
414 415
417 420
CHAPTER
XIII
CYCLIC SYSTEMS
174. 175. 176.
GENERAL EQUATIONS OF CYCLIC SYSTEMS CYCLIC CONGRUENCES SPHERICAL REPRESENTATION OF CYCLIC CONGRUENCES
426
431
.
432
CONTENTS
SECTION
177.
178.
XI
PAGE
436 437
439
179.
180.
SURFACES ORTHOGONAL TO A CYCLIC SYSTEM NORMAL CYCLIC CONGRUENCES CYCLIC SYSTEMS FOR WHICH THE ENVELOPE OF THE PLANES OF THE CIRCLES is A CURVE CYCLIC SYSTEMS FOR WHICH THE PLANES OF THE CIRCLES PASS THROUGH A POINT
440
CHAPTEE XIV
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
181.
TRIPLE SYSTEM OF SURFACES ASSOCIATED WITH A
CYCLIC
446
447 449
451
SYSTEM
182. 183.
184.
GENERAL EQUATIONS. THEOREM OF DUPIN
EQUATIONS OF LAME TRIPLE SYSTEMS CONTAINING ONE FAMILY OF SURFACES OF REVOLUTION TRIPLE SYSTEMS OF BIANCHI AND OF WEINGARTEN THEOREM OF RIBAUCOUR
185.
....
452
186.
187. 188.
THEOREMS OF DARBOUX TRANSFORMATION OF COMBESCURE
.
457 458
461
INDEX
467
DIFFERENTIAL GEOMETRY
CHAPTER
I
CURVES IN SPACE
1.
Parametric equations of a curve.
Consider space referred to
fixed rectangular axes, and let (x, y, z) denote as usual the coordi nates of a point with respect to these axes. In the plane 2 = draw a circle of radius r and center (a, b). The coordinates of a point P on the circle can be expressed in the form
(1)
x
a
{
r cos u,
y
= b H r sin u,
2
=
0,
where u denotes the angle which the radius to P makes with the to 360, the point P describes As u varies from positive
o&gt;axis.
the circle.
The
quantities
a, 5,
r determine the position
and
size
it.
of the circle,
whereas u determines the position of a point upon
a variable or parameter for the equations (1) are called parametric
it is is
In this sense
circle.
And
equations of the circle. straight line in space
A
determined by a
directioncosines
point on
a,
/3,
it,
P (a,
Q
6,
c),
and
its
7.
The
latter fix also the sense of the line.
Let
P
distance
be another point on the line, and let the P be denoted by u, which is positive Q
P
or negative. The rectangular coordinates of are then expressible in the form
(2)
P
x
= a + ua,
of
y
= b + u(B,
z
c
+ wy.
FIG. 1
To each value
u there corresponds a point on the line, and the coordinates of any point on the line are expressible as in (2). These equations are consequently parametric
equations of the straight line. When, as in fig. 1, a line segment
PD,
of constant length
,
per
pendicular to a line
OZ
at D, revolves uniformly about
OZ
as axis,
2
CURVES IN SPACE
and at the same time
locus of
D
moves along
it
P is
called a circular helix.
If
with uniform velocity, the the line OZ be taken for the
a&gt;axis,
2axis, the initial position of
PD for the positive
and the angle
between the
u,
latter
and a subsequent position
of
PD
be denoted by
the equations of the helix can be written in the parametric form
(3)
x
=a
cos u,
y
a sin u,
z
= bu,
where the constant
PD
and
of translation of D.
radian,
D
determined by the velocity of rotation of Thus, as the line PD describes a moves the distance b along OZ.
b is
the above equations u is the variable or parameter. Hence, with reference to the locus under consideration, the coordi indicate this by writing these nates are functions of u alone.
In
all of
We
equations
The functions / /2 / have
x,
,
definite
circle, straight line or circular helix.
eral case
and consider equations
(4),
forms when the locus is a But we proceed to the gen when /r /2 / are any func
,
tions whatever, analytic for all values of u, or at least for a certain domain.* The locus of the point whose coordinates are given by (4),
as
u takes
all
values in the domain considered,
is
a curve.
Equa
tions (4) are said to be the equations of the curve in the parametric all the points of the curve do not lie in the same plane form.
When
it is
called a space curve or a twisted curve ; otherwise, a plane curve. It is evident that a necessary and sufficient condition that a
curve, defined
linear relation
(5)
by equations (4), be plane, between the functions, such
ofi + 5f2 +
c/3
is
that there exist a
as
+d=
0,
where
dition
If
(6)
a, b, c,
is satisfied
d denote constants not all equal to zero. This con by equations (1) and (2), but not by (3). by any function of
*fc*^(*0,
v,
u
in (4) be replaced
say
equations
(4)
assume a new form,
*
E.g. in case
it is
u
is
when
complex,
it lies
two fixed values; supposed to be real, it lies on a segment between within a closed region in the plane of the complex variable.
EQUATIONS OF A CURVE
It is evident that the values of x, y,
z,
3
(7) for a
given by
value
are equal to those given by i u obtained from (6). Consequently equations (4) and (7) define the same curve, u and v being the respective parameters. Since
of of
,
(4) for the corresponding value
there
is
no restriction upon the function
&lt;/&gt;,
except that
it
be ana
lytic, it
follows that a curve can be given parametric representation
in
an infinity of ways. 2. Other forms of the equations of a curve.
If the first of
equa
tions (4) be solved for w, giving parameter, equations (7) are
(8)
u
$(#), then, in terms of x as
x
= x,
is
y
= F (x),
2
z
= F (x).
8
In this form the curve
or, if it
really defined
by the
its
last
two equations,
is
be a plane curve in the o?yplane,
equation
in the
customary form
(9)
y =/(*)
points in space whose coordinates satisfy the equation lie on the cylinder whose elements are parallel to the
The
z
y = F (x)
2axis
and whose cross section by the xypl&ne is the curve y = F2 (x). In like manner, the equation z = F3 (x) defines a cylinder whose
Hence the curve with the common to two cylinders equations (8) with perpendicular axes. Conversely, if lines are drawn through the points of a space curve normal to two planes perpendicular to one another, we obtain two such cylinders whose intersection is the given curve. Hence equations (8) furnish a perfectly gen
elements are parallel to the #axis.
is
the locus of points
eral definition of a space curve.
in
In general, the parameter u can be eliminated from equations (4) such a way that there result two equations, each of which in
all
volves
(10)
three rectangular coordinates.
Thus,
y, z)
Qfa
if
y, z)
=
0,
&lt;S&gt;
a (a;,
=
0.
Moreover,
the form
two equations
x,
of this kind be solved for
y and
z as
functions of
get equations of the form (8), and, in turn, of of u. Hence (4), by replacing x by an arbitrary function It will also are the general equations of a curve. equations (10)
we
be seen later that each of these equations defines a surface.
4
It
CURVES IN SPACE
should be remarked, however, that when a curve
is
defined
(10), it
as the intersection of
two cylinders
(8),
or of
two surfaces
may happen
that these curves of intersection consist of several
parts, so that the
new
equations define more than the original ones.
For example, the curve defined by the parametric equations
(i)
x
=
w,
y
=
w2
,
z
=
w3
,
is
a twisted cubic, for every plane meets the curve in three points. Thus, the plane
ax
+
by
f
cz
+ +
d
=
meets the curve in the three points whose parametric values are the roots of the
e(l uation
CM*
lies
+
&n"
+
an
d
=
0.
This cubic
upon the three cylinders
y
=
x2
,
z
=
x3
,
y3
=
z2
.
of the first
and second cylinders is a curve of the sixth degree, of the sixth degree, whereas the last two intersect in a curve of the ninth degree. Hence in every case the given cubic is only a part of
The
intersection of the first
and third
it is
the curve of intersection
Again,
(ii)
we may
eliminate
that part which lies on all three cylinders. u from equations (i), thus
xy
=
z,
y*
=
xz,
and the second a hyperboliclies on both of these surfaces, parabolic cone. The straight line y = 0, z = but not on the cylinder y = x 2 Hence the intersection of the surfaces (ii) consists
of
which the
first
defines a hyperbolic paraboloid
.
of this line
and the cubic. The generators of the paraboloid are defined by
x
=
a,
z
=
ay
6.
;
y
6,
z
=
bx
;
for all values of the constants a
and
From
(i)
we
see that the cubic meets each
generator of the
3.
is
first
family in one point and of the second family in two points.
Linear element.
By
the limit,
when
it exists,
definition the length of an arc of a curve toward which the perimeter of an
inscribed polygon tends as the
number
of sides increases
lengths uniformly approach zero. Curves for does not exist will be excluded from the subsequent discussion.
and their which such a limit
Consider the arc of a curve whose end points m mined by the parametric values U Q and # and let intermediate points with parametric values u^ w 2
, ,
,
ma are mv m
,
deter
,
2,
be
.
The length
l
k
of the chord
mkmk+l
is
=V2,r/;.^,
L1
)/v(oi
2
.
= i,
2,
3
LINEAR ELEMENT
By
the
5
is
mean value theorem
of the differential calculus this
equal to
where
f
.
t
= wt +
0&lt;(
w*+i ~ %)
differentiation.
&lt;
^
&lt;
*
and the primes indicate
,
As denned, the length of the arc m a is the limit of 2Z4 , as the lengths in k k+l tend to zero. From the definition of a definite
m
m
integral this limit is equal to
ra
n
)
Hence,
if s
denotes the length of the arc from a fixed point (u
to a variable point (u),
we have
This equation gives
(12)
s
as a function of w.
We
write
it
=(w),
(11) it follows that
and from
which we may write
(14)
in the
form
i
ds
ds
2
=dx
As thus expressed
is
called the element of length, or linear
element, of the curve.
In the preceding discussion we have tacitly assumed that u
real.
is
When
it is
complex we
take equation (11) as the definition
of the length of the arc.
If equation (12) be solved for u in terms of s, and the result be substituted in (4), the resulting equations also define the curve,
and
s is
the parameter.
From
(11) follows the
theorem
:
necessary and sufficient condition that the parameter arc measured from the point U = U Q is
2
A
u
be the
(15)
/,
+/r+/s = l2 2
An
exceptional case should be noted here, namely,
/r+/
"+/ a 2
=o.
6
CURVES IN SPACE
//
2, 3
Unless//,
be zero and the curve reduce to a point, at least one
of the coordinates
must be imaginary. For
this case s is zero.
Hence
these imaginary curves are called curves of length zero, or minimal curves. For the present they will be excluded from the discussion. Let the arc be the parameter of a given curve and s and s + e
its
values for two points M(x,
y, z)
and M^(x^ y^ z^.
By Taylor s
theorem we have
(17)
^
=z
f z e
where an accent indicates
Unless x
1
differentiation with respect to
is,
s.
,
y
,
z
!
are all zero, that
unless the locus
is
a point
z is zl and not a curve, one at least of the lengths x l x, If these lengths be denoted by of the order of magnitude of e. &u,
y^y,
%,
Sz,
and
e
by 8, then we have
denotes the aggregate of terms of the second and higher the ratio of the lengths orders in 8s. Hence, as l approaches limit of the chord and the arc approaches unity and in the
where
1
2
M
M
MM
.
l
;
we have
4.
ds
2
=
2 dx2 + dy 2 + dz
is tangent to a curve at a point of a point and l the limiting position of the secant through as a limit. the curve as the latter approaches In order to find the equation of the tangent we take s for par in the 1 ameter and write the expressions for the coordinates of are and The equations of the secant through l form
Tangent
to
a curve.
The
M
M M
M
(17).
M
M M
of these equations be multiplied by e and the denominators be replaced by their values from (17), we have in
If
each
member
the limit as
M
1
approaches
M
y
TANGENT TO A CURVE
If #,
/3,
7
7 denote the directioncosines of the tangent in conse
quence of (15),
we may take
u
* any whatever, these equations are
f! /2
.
When
* (20) 9
the parameter
ft Jl
/
is
a=
nt
*i
^
*
o
=,
.
/wo
0=
/
/*o
.
/*&gt;.
=
=
&gt;
/!*)
y
*
= =
//
ft
3
They may
/oi\
also be written thus
:
21)
a
dx = :T
ds
=
dy
ds
/
dz
T"
V
ds
From these equations it follows that, if the convention be made that the positive direction on the curve is that in which the par ameter increases, the positive direction upon the tangent is the
same
as
upon the curve.
fundamental property of the tangent is discovered by con sidering the expression for the distance from the point M^ with
the coordinates (17), to any line through M. tion of such a line in the form
A
We
write the equa
(22)
*=
a
a, 5, c
= !=* =
b
=,
c
where
are the directioncosines.
The
(23)
distance from
{[(bx&gt;
M
l
to this line is equal to
\(bx" ay")e*
ay
)e
+
+
1
2
]
2
.
bz )e
+
2
]
+ [(az  cx )e +
]
}*.
Hence,
if
MM
l
be considered an infinitesimal of the
first order,
this distance also is of the first order unless
in
which case
it is
of the second order at least.
But when
these
equations are satisfied, equations (22) define the tangent at M. Therefore, of all the lines through a point of a curve the tangent
is
nearest to the curve.
,
,
* Whenever the functions x y z appear in a formula it is understood that the arc s is the parameter otherwise we use /{, /2 /3 indicating by accents derivatives with respect to the argument u.
;
,
,
8
5.
CURVES IN SPACE
Order of contact.
Normal plane.
When
the curve
is
such
that there are points for which
(24)
^=4
x
y
z
the distance from
to the tangent is of the third order at least. l In this case the tangent is said to have contact of the second order, whereas, ordinarily, the contact is of the first order. And, in gen eral, the tangent to a curve has contact of the wth order at a point,
M
if
the following conditions are satisfied for n
=
2,
,
n
1,
and n
:
&lt;
25 )
^=^=^rr
Jl
fl
~jTf
xw
i(0 jyV V
5&gt;V"y
the parameter of the curve is any whatever, equations (24), (25) are reducible to the respective equations
f (rt1)
2,
When
J%
J%
//
J\
f(.n\)
J
Ja
f(nl)
The plane normal
contact
(26)
is
to the tangent to a curve at the point of normal plane at the point. Its equation is called the
(X
a,
/3,
x)
a
+ (Y
y}
ft
+ (Z
z)
7
=
0,
where
7 have the values
(20).
EXAMPLES
1.
Put the equations
of the circular helix (3) in the
form
(8).
Express the equations of the circular helix in terms of the arc measured from a point of the curve, and show that the tangents to the curve meet the elements of the circular cylinder under constant angle.
2.
3.
Show
is
that
if
at every point of a curve the tangency
line.
is
of the second order,
the curve
4.
a straight
sufficient condition that at the point (x 2/o) of = / "(BO) f(x) the tangent has contact of the nth order is/"(x ) the tangent crosses the = also, that according as n is even or odd r=/()(z ) = curve at the point or does not.
,
Prove that a necessary and
the plane curve y
. . .
;
Prove the following properties of the twisted cubic the cubic one and only one meets the (a) Of all the planes through a point of 3 cubic in three coincident points its equation is 3 u*x  3 uy + z  w = 0. on a plane has a the orthogonal projection (6) There are no double points, but
5.
: ;
double point.
a variable chord of the cubic and by each of (c) Four planes determined by four fixed points of the curve are in constant crossratio.
FIKST
6.
CUKVATUKE
first
9
Let Jf, be two l points of a curve, As the length of the arc between these points, and A0 the angle between the tangents. The limiting value of
Curvature.
Radius of
curvature.
M
A0/As
as
M
l
approaches
Jf,
namely dd/ds, measures the
rate of
as the point of con change of the direction of the tangent at tact moves along the curve. This limiting value is called the
its reciprocal the radius of the latter will be denoted by p. first In order to find an expression for p in terms of the quantities defining the curve, we introduce the idea of spherical representa
first
M
curvature of the curve at M, and
;
curvature
take the sphere * of unit radius with center at the origin and draw radii parallel to the positive directions ofthe tangents to the curve, or such a portion of it that no two
tion as follows.
We
tangents are parallel. The locus of the extremities is a curve upon the sphere, which is in onetoone correspondence with the
given curve.
In this sense
we have
a spherical representation, or
spherical indicatrix, of the curve.
The angle A# between the tangents to the curve at the points M, M^ is measured by the arc of the great circle between their
on the sphere. If ACT denotes the representative points m, l of the arc of the spherical indicatrix between and m^ length then by the result at the close of 3,
m
m
dO
=v lim
!
p
A&lt;9
= 1.
da
Ao
Hence we have
(27)
=
ds
,
where
da is
The
the linear element of the spherical indicatrix. coordinates of are the directioncosines a, /3, 7 of the
m
tangent at
M\
consequently
When
(28)
the arc
s is
the parameter, this formula becomes
^.jji+yw+gw
* Hereafter
we
refer to this as the unit sphere.
10
CURVES IN SPACE
is
However, when the parameter from (12), (13), (20),
any whatever,
u,
we have
and
Wfifl+JSJf*flfr
find
Hence we
(30)
by substitution
which sometimes
is
written thus
:
\ _(d*x}*
fmake sign of p is not determined by these formulas. the convention that it is always positive and thus fix the sense of
The
We
a displacement on the spherical indicatrix.
7.
Osculating plane.
and through a point M^ of the curve. The a curve at a point is called the of this plane as approaches l limiting position and thus establish osculating plane at M. In deriving its equation
M
Consider the plane through the tangent to
M
M
ing its existence we assume that the arc s is the parameter, and in the form (17). take the coordinates of l
M
The equation
(32)
of a plane through
M
(x, y, z) is
of the
form
X,
Y",
Z
being
(X x)a + (Y y)l + (Zz)c = Q, the current coordinates. When the
Jf,
through the tangent at
(33)
If the values (17) for
the coefficients
a,
c
&gt;,
plane passes are such that
xa
a;
,
+yb+ zc=
y^
z^
0.
be substituted in (32) for X, F, Z,
e*
,
and the resulting equation be divided by
we
get
where
limit
(34)
77
represents the aggregate of the terms of
first
orders in
As e. we have
M
1
approaches
Jf,
77
approaches zero,
and higher and in the
x"a
+
y"b
+
z"c
=
0.
OSCULATING PLANE
Eliminating
a,
11
c
&gt;,
from equations
(32), (33), (34)
we
obtain, as
the equation of the osculating plane,
Xx Yy Zz
(35)
x
1
y
y"
=
0.
x"
From
plane
(36)
this
we
find that
when
the curve
w,
in terms of a general
is
parameter
is defined by equations (4) the equation of the osculating
x_ x Y
//
y
/,"
z_ z
fi
The plane defined by either of these equations is unique except when the tangent at the point has contact of an order higher than
the
In the latter case equations (33), (34) are not independent, as follows from (24); and if the contact of the tangent is of the wth order, the equations =
first.
^+^+^
~\.,
G
0?
for all values of r
one another.
up to and including n are not independent of this equation and (33) are inde But for r = n + pendent, and we have as the equation of the osculating plane at this
singular point,
Xx Yx
Zz
=
0.
y
plane, and its plane is taken for the rryplane, reduces to Z = 0. Hence the osculating plane the equation (35) of a plane curve is the plane of the latter, and consequently is the same for all points of the curve. Conversely, when the osculating
When
a curve
is
plane of a curve is the same for all its points, the curve is plane, for all the points of the curve lie in the fixed osculating plane.
The equation
reducible to
of the osculating plane of
the twisted cubic
(2)
is
readily
where JT, F, Z are current coordinates. From the definition of the osculating plane and the fact that the curve is a cubic, it follows that the osculating plane meets
the curve only at the point of osculation. As equation (i) is a cubic in w, it follows that through a point (o, 2/o, ZQ) not on tne cnrve there pass three planes which osculate the cubic. Let MI, w 2 u 3 denote the parameter values of these points.
,
Then from
(i)
we have
=
3
XG,
3
?/o,
2t
i
\
n
12
By means
CURVES IN SPACE
of these relations the equation of the plane through the corresponding is reducible to
three points on the cubic
(X 
XQ) 3
7/0
 Y(
y
,
)
3x
+
ZQ)
5
(Z

z
)
= 0.
:
This plane passes through the point
(x
2/0,
hence we have the theorems
The points of contact of the three osculating planes of a twisted cubic through a point not on the curve lie in a plane through the point. The osculating planes at three points of a twisted cubic meet in a point which lies
in the plane of the three points.
By means of these theorems we can establish a dual relation in space by mak ing a point correspond to the plane through the points of osculation of the three osculating planes through the point, and a plane to the point of intersection of the three planes which osculate the cubic at the points where it is met by the plane.
In particular, to a point on the cubic corresponds the osculating plane at the point,
and
vice versa.
8.
Principal normal and binormal.
Evidently there are an in
finity of
normals to a curve at a point.
:
Two
of these are of par
ticular interest
the normal, which lies in the osculating plane at the point, called the principal normal; and the normal, which is perpendicular to this plane, called the binormal.
If the directioncosines of the
binormal be denoted by X,
/&gt;t,
z&gt;,
we have from
X
:
(35)
:
/
v
= (y
z"
z
y")
:
(z
x"~
z
:
z")
(*
/ y
x").
In consequence of the identity
the value of the
(28) to
common
ratio
is
reducible by means of (19) and
take the positive direction of the binormal to p.* be such that this ratio shall" be f p then
;
We
(37)
\ = p(y z"z
f
y"),
^ = P (z
is
x"~
x
z"),
v
= p(x y
When
(38)
the parameter u
general, these formulas are
x=
282
:
or in other form
/oof
~ P dycPz
*For
/
dzcfy
~~df~
0,
^~ P
~
dzd*xdxd?z
ds
3
~ P dxd y
1
dyd^x
ds*
SV =
as
is
seen by differentiating
2x"*=
with respect to
s.
PRINCIPAL NORMAL AND BINORMAL
By
definition the principal
13
normal
tangent and binormal. We make direction is such that the positive directions of the tangent, prin cipal normal and binormal at a point have the same mutual ori
entation as the positive directions of the x, y, zaxes respectively.
perpendicular to both the the convention that its positive
is
These directions are represented in
fig.
2 by the lines
if
Z,
MT, MC, MB.
Hence,
m,
n,
denote the direc
tioncosines of the principal normal,
we have*
(39)
m
/JL
=41,
FIG. 2
from which
it
follows that
I
= mv = \ = ftn
a
I fjij
,
nu,
vft^
777*,
ft
n\
va
yl
Iv,
7
m=
/Ji
\7,
#n,
n
i^
=
= Z/i = Xp = am
m\,
yuo:,
ftl.
Substituting the values of #, /3, 7; X, /u, i^ from (19) and (37) in the m, w, the resulting equations are reducible to expressions for
,
Hence, when the parameter u
(42)
is
general,
we have
l=(W2
or in other form,
_
In consequence of (29) equations (42)
2
dzd 2 s
may
be written:
da
(43)
dft
dj
ds
ds
(27),
ds
or
by means of
da
m
dft
do
dy
da
Hence the tangent
to the principal
to the spherical indicatrix of a curve is parallel
normal to the curve and has the same sense.
*C. Smith, Solid Geometry, llth
ed., p. 31.
14
9.
CURVES IN SPACE
Osculating
circle.
Center of
first
curvature.
We
have defined
the osculating plane to a curve at a point to be the limiting of the plane determined by the tangent at and by a position of the curve, as the latter approaches the curve. point l along
M
M
M
M
We
consider
now
gent at
to the
tion of this circle, as
M as the curve, and passes through M The limiting posi M approaches called the osculating
{
.
the circle in this plane which has the same tan
l
7I/,
is
circle
curve at M.
It is evident that its center
C
is
on the prin
normal at M. Hence, with reference F the coordinates of (7 denoted by Q
cipal
,
to
X
,
,
Z
Q,
any fixed axes in space, are of the form
rn,
X =x +
rl,
Y
y
f
rm,
Z^=z +
where the absolute value of
of the circle,
r is the radius of the osculating circle.
r,
In order to find the value of
we
return to the consideration
its
when
M
l
does not have
limiting position, and
we
let X, F, Z\ Zj, m^ n^ r^ denote respectively coordinates of the cen ter of the circle, the directioncosines of the diameter through and the radius. If x v y v z l be the coordinates of M^ they have the
M
values (17), and since
rl
If
M
l
is
on the
circle,
we have
i
2
e*x".
= 2(A  xtf = 2(7^ ex 2
)
.
we
divide
^x ^ = 0, and through by e we have
notice that
,
after reducing the
above equation
1r^Z/
where
77
+*?
=
(),
involves terms of the
limit r l
becomes
r,
^x\
and higher orders in e. In the becomes 2z 7, that is  and this equation
first
,
reduces to
so that r
is
equal to the radius of curvature.
circle
On
this
account the
of curvature and its center the osculating circle is called the center of first curvature for the point. Since r is positive the center
of curvature
is
on the positive half of the principal normal, and
coordinates are
pl,
consequently
(44)
its
X =x +
Y =y + pm,
Z = z + pn.
Q
CENTER OF CURVATURE
The
ture
is
15
line
normal to the osculating plane at the center of curva
called the polar line or polar of the curve for the corre sponding point. Its equations are
/45\
Xxpl = Yypm = Zz
\
fig.
JJL
pn
^
v
In
2
C
represents the center of curvature and
CP
the polar
line for
M.
A curve may be looked upon as the path of a point moving under the action of a system of forces. From this point of view it is convenient to take for parameter the time which has elapsed since the point passed a given position. Let t denote this parameter. As t is a function of 8, we have
dx
_dx ~
ds
_
ds
C
dy
ds~dt~ *dt
Hence the
_ dt~
ds
~dt
dz
_
ds
~dt~
y
~dt
rate of change of the position of the point with the time, or its velocity,
laid off dt
may
be represented by the length
(41),
on the tangent
to the curve.
In like
manner, by means of
we have
d^ _ ~
From
this it is seen that the rate of
d?s
n
/ds\ 2
7
change of the velocity at a point, or the
acceleration,
be represented by a vector in the osculating plane at the point, through the latter and whose components on the tangent and principal normal
may
d*s

, and 
1
/dY
I
)
df*
P \dtj
EXAMPLES
1.
Prove that the curvature of a plane curve defined by the equation
M
(x,
y)dx
cy
p
2.
ex
(J/
2
+
N
z
Show
that the normal planes to the curve,
x
a sin 2
it,
y
=
a sin u cos w,
=
a cos M,
pass through the origin, and find the spherical indicatrix of the curve.
3.
The
straight line
is
the only real curve of zero curvature at every point.
:
4.
(a)
Derive the following properties of the twisted cubic
is
In any plane there planes can be drawn.
(6)
one
line,
and only one, through which two osculating
fixed osculating planes are cut by the line of intersection of any two osculating planes in four points whose crossratio is constant. and four fixed points of the curve (c) Four planes through a variable tangent
Four
are in constant crossratio.
(d)
What
is
the dual of
(c)
by the
results of
7?
16
5.
CURVES IN SPACE
Determine the form of the function
curve x
6.
=
w,
y
=
sin w, z
&lt;f&gt;
so that the principal normals to the
(u) are parallel to the yzplane.
first
Find the osculating plane and radius of x a cos u f 6 sin w, y = a sin u
curvature of
z
+
6 cos w,
=
c sin 2 u.
10. Torsion. FrenetSerret formulas.
less a
It has
been seen
that,
un
curve be plane, the osculating plane varies as the point
moves along the curve. The change in the direction depends evidently upon the form of the curve. The ratio of the angle A^ between the binormals at two points of the curve and their curvi linear distance As expresses our idea of the mean change in the
direction of the osculating plane.
this ratio, as
And
so
we take
the limit of
one point approaches the other, as the measure of rate of this change at the latter point. This limit is called the
the second curvature, or torsion, of the curve, and
its
inverse the
radius of second curvature, or the radius of torsion. will be denoted by r.
The
latter
In order to establish the existence of this limit and to find an
expression for
it
in terms of the functions defining the curve,
we draw
radii of the unit sphere parallel to the positive binormals
of the curve
and take the locus of the end points
of these radii as
a second spherical representation of the curve.
The
coordinates of
points of this representative curve on the sphere are X, /*, v. Pro ceeding in a manner similar to that in 6, we obtain the equation
(46)
i_
r
dcr l is
2
ds*
where
the linear element of the spherical indicatrix of the
binormals.
In order that a real curve have zero torsion at every point, the cosines
X,
/*,
v
must be constant. By a change of the fixed axes, which evidently has no effect upon the form of the curve, the cosines can be given the values X = 1, /* = v = 0. const. Hence a necessary It follows from (40) that a = 0, and consequently x and sufficient condition that the torsion of a real curve be zero at every point is
that the curve be plane.
In the subsequent discussion
we
shall
need the derivatives with
7;
I,
respect to s of the directioncosines a,
&
m, w;
X, p, v.
We
deduce them now.
(4T)
From
a
(41)
/3
we have
=i,
=, y.
FRENETSERRET FORMULAS
In order to find the values of X
respect to
s
,
17
/*
,
i/,
we
differentiate with
the identities,
X 2 +At2+z 2 = 1?
,
a\
+
/A
+
7*
= 0,
r
and, in consequence of (47), obtain
XX
+
1
fjLfj,
+
vv
1
= 0,
f :
\
+ /V + yv =
0.
From
and
these,
by
the proportion (40), follows
X
fjL
:v
r
=l:m:n,
is
the factor of proportionality
algebraic sign of r
its
is
1/r, as is seen
from
(46).
The
We fix
(48)
not determined by thus sign by writing the above proportion
the latter equation.
:
V= A
T
I
,:
&gt;
=
T
t
V
=
T
s
l*&gt;
If the identity
=
^7
vfi
be differentiated with respect to
the result
(49)
is
reducible by (40), (47), and (48) to
I
and n Similar expressions can be found for Gathering to fundamental in gether these results, we have the following formulas
.
m
the theory of twisted curves, and called the FrenetSerret formulas
:
(50)
\^ v v ^/l+2),
Y
As an example, we
If the equation
derive another expression for the torsion.
\
= p(y z
"
z
y")
be differentiated with respect to
s,
the result
may be
written
and similar ones for WI/T, rz/r be multiplied by ?, wz, n respectively and added, we have, in consequence of (50) and (41),
If this equation
x
(51)
y
y"
" "
z
z"
x
y
z
"
18
CUKVES IN SPACE
The last three of equations (50) give the rate of change of the directioncosines of the osculating plane of a curve as the point of osculation moves along the curve. From these equations it follows
that a necessary and sufficient condition that this rate of change at a point be zero is that the values of s for the point make the determinant in equation (51) vanish. At such a point the osculat
ing plane
11.
torsion.
is
said to be stationary.
of curve in the neighborhood of a point. The sign of have made the convention that the positive directions of the tangent, principal normal, and binormal shall have the same
Form
We
relative orientation as the fixed
take these lines at a point for axes, the equations of the curve Q can be put in a very convenient form. If the coordinates be ex
M
x, y,
2axes respectively.
When we
and
pressed in terms of the arc measured from = (41) that for s
M^ we
P
have from (19)
When
the values of
I
and X from
(41)
and
(37) are substituted in
the fourth of equations (50),
(5 2)
we
r
obtain
x
this
"
=  (y
p
z"
z
y")
"
x".
p
for
f
From
for
and similar expressions
i
y
and
i
1
z"
we
find that
s=
2
P*
p
PT
#, y, z
Hence, by Maclauriri pressed in the form
s
/
theorem, the coordinates
i
can be ex
(^%\
I
\&lt;JO
&lt;
v v
z
=
Ct
2p
bp
s
J
2
s
3
f,
f
,
6 pr
where p and r are the radii of first and second curvature at the = 0, and the unwritten terms are of the fourth and higher point s
powers
in
s.
the last of these equations it is seen that for sufficiently small values of 8 the sign of z changes with the sign of s unless
From
THE SIGN OF TORSION
I/T
19
Hence, unless the osculating plane is stationary at the curve crosses the plane at the point.* Furthermore, a point, when a point moves along a curve in the positive direction, it side of the osculating passes from the positive to the negative at a point, or vice versa, according as the torsion at the
at
.
=
M
plane
or negative. In the former case the curve said to be sinistrorsum, in the latter dextrorsum.
latter is positive
is
As another consequence
variable point
to the osculating plane at
M on the curve approaches Jf the distance from M M of the third order of magnitude in
,
of this equation,
we remark
that as a
Q
is
comparison with
MM
Q
.
By means
any
find that the distance to
of the other equations (53) we is of the other plane through
M
:
second order at most.
Hence we have the theorem
to
The osculating plane
crossed by the curve,
a twisted curve at an ordinary point
all the
is
and of
planes through the point
it lies
nearest
to the
curve.
it
is positive for suffi or negative. Hence, in the positive neighborhood of an ordinary point, the curve lies entirely on one on side of the plane determined by the tangent and binomial
From
the second of (53)
,
is
seen that y
ciently small values of
the side of the positive direction of the principal normal. These properties of a twisted curve are discovered, likewise,
from a consideration of the projections upon the coordinate planes of the approximate curve, whose equations consist of the first
terms in
to the
(53).
The
y=
parabola x
the
cubic
= *,
x
curve.
On
s,
the projection on the osculating plane is whose axis is the principal normal /2 p, the plane of the tangent and binomial it is
s
2
=
z
=
s
8
/6 pr,
which has the
curve for an inflectional tangent. And the plane of the binormal and principal normal into the semi3 2 with the latter for s /Q pr, cubical parabola y = s /2 /o, z=
cuspidal tangent.
These results are represented by the following figures, which picture the pro upon the osculating plane, normal plane, and the plane of the
In the third figure the heavy line corresponds to the case
line to the case
is
tangent to the the curve projects upon
jection of the curve
tangent and binormal.
where r
positive
and the dotted
where r
is
negative.
*This result can be derived readily by geometrical considerations.
20
CUEVES IN SPACE
The preceding results serve also to give a means of determining; ue variation in the osculating plane as the point moves along the curve. By r ns of (50) the directioncosines X, /u,, v can be given the form
where the subscript null indicates the value of a function for s = and the un written terms are of the second and higher terms in s. If the coordinate axes are those which lead to (53), the values of X, p, v for the point of parameter 5s are
X
=
.
=,
TO
v
=
\
to within terms of higher order,
and consequently the equation of
this osculating
plane at this point
MI
is
Y + Z =
TO
;
0.
M
we put = po, we get the zcoordinate of the point in which this plane is cut by it is the polar line for the point s = po5s/T Hence, according as T O is positive or negative at Jf, the osculating plane at the nearby point MI cuts the polar line for on the negative or positive side of the osculating plane at
If
Y
.
M
.
12. Cylindrical helices.
of the use of formulas (50) we derive several properties of cylindrical helices. By definition, a cylindrical helix is a curve which lies upon a cylinder and cuts the elements of the cylinder under constant angle. If the axis of
As another example
z be taken parallel to the elements of the cylinder,
we have 7 =
const.
Hence,
from
(50),
from which
it
follows that the cylindrical helices have the following properties
:
The principal normal is perpendicular to the element of the cylinder and consequently coincides with the normal to the cylinder at the point ( The radii offirst and second curvature are in constant ratio.
:
at the point,
22).
Bertrand has established the converse theorem Every curve whose radii of first and second curvature are in constant ratio is a cylindrical helix. In order to prove = icp, and remark from (50) that it, we put T
da_d\
ds ds K\
dp
_
*/*
dfj.
dy
ds
_ =
KV
dv
ds
ds~
/3
ds
from which we get
where
a2
a
+
C2
a,
=
+
6,
7
+
cy
c,
a, 6, c are constants.
From
these equations
we
find
+
52
+
_
i
+ K^
aa + bp +
=
CYLINDRICAL HELICES
Hence the
lines
ta
21
ents to the
curve
make
V
*
the constant angle cos *
=;
is
with the
whose
lin
3ncosines are
ts
Vl +
helix,
Consequently the curve
K2
a cylindrical
and
t .u
e
of the helix have the above direction.
EXAMPLES
sin w), y = a cos w, between the points 1. Find the length of the curve x = a (u show that the locus of the center of curva TT and IT for which u has the values ture is of the same form as the given curve.
;
2.
Find the coordinates of the center of curvature of
x
=
a cos M,
y
=
y
a sin
w,
z
a cos 2
u.
3.
Find the
radii of curvature
and torsion of
x
4.
is
=
a (u
sinw),
=
a
(1
cosw),
z
=
bu.
If the principal
normals of a curve are parallel to a fixed plane, the curve
a cylindrical helix.
,
V% u is a cylindrical helix and that e u y = er M , z 5. Show that the curve x the right section of the cylinder is a catenary also that the curve lies upon a cylin der whose right section is an equilateral hyperbola. Express the coordinates in terms of the arc and find the radii of first and second curvature.
;
6.
Show
that
if
6
and
curve
7.
first
make with
a fixed line in space, then
denote the angles which the tangent and binormal to a sin 6 dd =r

sin
&lt;/&gt;
d&lt;p
p
When two
curves are symmetric with respect to the origin, their radii of
curvature are equal and their radii of torsion differ only in sign.
8. The osculating circle at an ordinary point of a curve has contact of the sec ond order with the latter and all other circles which lie in the osculating plane and are tangent to the curve at the point have contact of the first order.
;
9. A necessary and sufficient condition that the osculating circle at a point have contact of the third order is that p = and I/T = at the point at such a point
;
the circle
10.
is
said to superosculate the curve.
Show
that any twisted curve
may
be defined by equations of the form
where
11.
p
and r are the
radii of first
and second curvature at the point
(4),
s
is
0.
When
the equations of a curve are in the form
/I
f
the torsion
given by
/2
fff
/3
f/f
f
where
has the significance of equation
(12).
22
12.
CURVES IN SPACE
The locus
is
of the centers of curvature of a twisted curve of constant first
curvature
13.
a curve of the same kind.
all
When
is
the osculating planes of a curve pass through a fixed point, the
curve
14.
plane.
plane.
Determine f(u) so that the curve x What is the form of the curve ?
=
a cosw, y
=
a sin w,
z
=f(u)
shall be
two curves defined
Fundamental theorem. Let C^ and Cz be s, and let points each with the same values of s correspond. We assume, upon
13. Intrinsic equations.
in terms of their respective arcs
ture have the same value,
furthermore, that at corresponding points the radii of first curva and also the radii of second curvature.
We
shall
show that Cl and Cz
are congruent.
By a motion in space the points of the two curves for which = can be made to coincide in such a way that the tangents, s principal normals, and binomials to them at the point coincide
also.
Hence
s
indicate
use the notation of the preceding sections and by subscripts 1 and 2 the functions of Cl and C2 we have,
if
,
we
when
(54)
=
0,
xl
=x
z,
al
=a
z,
^
=
Z
2,
\=\
z,
and other similar equations.
The
FrenetSerret formulas for the two curves are
ds
ds
r
ds
=ds
=
=
p
&gt;
=
ds
ds
(


I
=
r
&gt;
\p
T/
the functions without subscripts being the same for both curves. If the equations of the first row be multiplied by 2 Z 2 X 2 respec tively, and of the second row by a^ l^ X : and all added, we have
,
,
,
(55)
and consequently
This constant
is
^(,.H*,+ i\)=0. a^ + IJ + \\ = const.
2
x
equal to unity for
s
= 0,
as is seen
from
(54),
and
hence
for all values of s
we have
INTRINSIC EQUATIONS
Combining
this equation
23
with the identities
we
obtain
(a t
 a + ft 2
2)
2
&gt;)
+ (X,  X = 0.
2
2)
Hence a
&=&
= a# ^ = X = X = 7 we nave 7i
l
Z
2,
t
2
.
Moreover, since in like manner
2
1(^^=0,
But
^^=0, I
,
&lt;*,,)
= o.
Consequently the differences 2^ #2 y^y^ z l z 2 are constant. for s = they are zero, and so we have the theorem
:
curves whose radii of first and second curvature are the same functions of the arc are congruent.
Two
From
this it follows that a
curve
is
determined, to within
its
position in space, by the expressions for the radii of first and second curvature in terms of the arc. And so the equations of a curve
may be
(56)
written in the form
/&gt;=/,&lt;),
T =/,(.).
They
are called its intrinsic equations.
inquire, conversely, whether two equations (56), in which f^ are any functions whatever of a parameter s, are intrinsic
We
and/
2
equations of a curve for which s is the length of arc. In answering this question we show, in the first place, that the
equations
/trrv
(pi)
du ds
=v ,
p
dv
ds
=
/u
/
w\
,
dw
ds
v
__
\p
T/
:
r
admit of three sets of solutions, namely
(58)
u
= a,
;
v
=
1,
w=
\; u
= fi,
v
= m, w =
/JL
;
u
y^ v
= n, w = v;
which are such that for each value of s the quantities a, fi, 7; v are the directioncosines of three /, 7?z, n X, mutually perpen dicular lines. In fact, we know * that a system (57) admits of a set of solutions whose values for s = are given arbitra unique
/i,
rily.
Consequently these equations admit of three sets of solutions
II, p.
*
Vol.
Picard, Tralte d Analyse, Vol.
II, p. 356.
313; Goursat, Cours
d Analyse Mathematique,
24
CURVES IN SPACE
s
whose values for
=
s
are 1, 0,
;
0, 1,
;
0, 0,
1 respectively.
By
(59)
an argument similar to that applied to equation (55) we prove
the solutions (58) satisfy the conditions
0,
it
that for all values of
aft
+ Im + \p =
7
+ mn + pv = 0,
dv
ya
+
nl
+ v\ =
0.
In like manner, since
follows from (57) that
ur+v as + w as
we prove
(
du
dw =
as
A
(),
that these solutions satisfy the conditions
a
60)
o
+Z
a
+X = l,
a
2
/3
+m
2
+/x
2
= l,
y+n
+i/^l.
(59), (60) are equivalent to (40), and conse 7; Z, TH, w; X, /z, v are quently the three sets of functions a, the directioncosines of three mutually perpendicular lines for all values of s.
But the conditions
&
Suppose we have such a set of solutions. For the curve
(61)
x
I
ads,
y=
,
I
fids,
z=*
I
yd*,
the functions
since ds*
(61)
=
a, /3, 7 are the directioncosines of the tangent, and dx 2 + dy* + dz 2 s measures the arc of the curve. From
first
and the
of (57)
we get
d?x_l_
ofy^m ~
ds
2
ds*~p
p*
d^z^n. df~~p
/d*x\*
/^&gt;\
2
W/ W/ W/
/^\ = 1
p*
2
Hence if p be positive for all values of s, it is the radius of curva ture of the curve (61), and Z, m, n are the directioncosines of the principal normal in the positive sense. In consequence of (40) the
functions X,
yu.,
v are the directioncosines of the binomial; hence of (57) it follows that r
is
from
(50)
and the third
the radius of
torsion of the curve.
Therefore we have the following theorem in the theory of curves fundamental
:
is
Given any two analytic functions, f^s), f2 (s), of which the former a positive for all values of s within a certain domain ; there exists
and s is the arc, for values of s in curve for which p =/j(s), r 2 (), the given domain. The determination of the curve reduces to the find
ing of three sets of solutions of equations (57), satisfying the conditions
(59), (60),
=/
and
to
quadratures.
KICCATI EQUATIONS
25
We proceed now to the integration of
set of integrals of the desired kind
(62)
must
equations (57). Since each satisfy the relation
u2
cr
we
* two functions introduce with Darboux
and
&),
defined by
1
(63)
w
iv
u
1
iv
u
\
w
u
cr
+w + iv
and
ft)
It is
evident that the functions
v,

are conjugate imaginaries.
Solving for u,
(64)
w,
we
1
get
v
u
=1
oft)
=
,1
i
+
oft)
i
w=
cr
+
co
If these values
be substituted in equations (57),
it is
found that
the functions
cr
and
co
are solutions of the equation
(65)
Miieip.
ds
2r
p
2r
when substi conversely, any two different solutions of (65), tuted in (64), lead to a set of solutions of equations (57) satisfying the relation (62). Our problem reduces then to the integration of
And
equation (65).
14.
Riccati equations.
Equation (65)
2
may be
written
(66)
^=L+
N
for him.
MO + NP,
are functions of s. This equation is a generalized where L, M, form of an equation first studied by Riccati, f and consequently
is
named
As
theory of curves
properties.
and
Riccati equations occur frequently in the surfaces, we shall establish several of their
a particular integral of a Riccati equation known, the general integral can be obtained by two quadratures.
Theorem.
When
is
*
treatise frequently,
t
.
Lemons sur la Thdorie Generate des Surfaces, Vol. I, p. 22. We shall refer to this and for brevity give our references the form Darboux, I, 22. Cf Forsyth, Differential Equations, chap, v also Cohen, Differential Equations,
;
pp. 173177.
26
CURVES IN SPACE
we put
Let O l be a particular integral of (66). If is the equation for the determination of
&lt;/&gt;
= l/c
f
0^

(67)
+ 2(M+2WJ&lt;l&gt;+N**Q.
and of the
first
As
this equation is linear
order,
it
can be solved by
two quadratures.
&lt;=/1 (s)+ 0/2(3), general integral of equation (66)
Since the general integral of (67) is of the form where a denotes the constant of integration, the
is
of the
form
Sriwhere P,
(),
R,
S
are functions of
s.
Theorem. When two particular integrals of a Riccati equation are known, the general integral can be found by one quadrature.
Let
l
and
2
be two solutions of equation (66).
If
we
effect the
substitution 6
\6
,
the equation in
^
is
respec equation and (67) be multiplied by 1/^r and and subtracted, the resulting equation is reducible to tively,
If this
!/&lt;
ty/&lt;l&gt;)=N(0 1
2 )^/&lt;t&gt;.
Consequently the general integral of
cts
(66)
is
given by
69
&lt; &gt;
00,
vr
fv(0 t 0c/
where a
is the constant of integration. Since equation (68) may be looked upon as a linear fractional corre substitution upon a, four particular solutions V 2 # 3 4
,
, ,
sponding
to four values a v a z , a 3 a
,
of
a,
are in the
:
same
crossratio
as these constants.
Hence we have the theorem
The
crossratio
is
of any four particular integrals of a Riccati
equation
constant.
From
this it follows that if three particular integrals are
known,
the general integral can be obtained without quadrature.
DETERMINATION OF COORDINATES
15.
27
The determination
equations.
of the coordinates of a curve defined
by
its intrinsic
We return to
(o
i
the consideration of equation
(65)
(70)
and indicate by
o\
= a.P+O ^ ^i
= bP+Q v
T&gt;
(i
= l,2,3)
we
obtain
six particular integrals of this equation. three sets of solutions of equations (57),
*1
I
From
namely
these
(71)
for /8, m, ft; and similar expressions in cr 2 3 respectively 3 2 These expressions satisfy the conditions (60). In order 7, n, ZA that (59) also may be satisfied we must have
,
&&gt;
;
&lt;7
,
&&gt;
CT
ft),,
ft),
ft),
which
(72)
is
reducible to
= 1.
of the three pairs of constants
z.
A/
_1
;
JL
9
^
^
q O
*
^
,
Hence each two
5X
2,
J2
;
a3
form a harmonic range.
When the
for a,
/3,
7,
values (70) for it is found that
&)
&lt;T.,
.
t
are substituted in the expressions
(73)
7
=
,1+a,
a
where, for the sake of brevity,
we have put
(74)
RS
PQ
PSQR
28
CURVES IN SPACE
The
coefficients of U, F,
and
W in
,
(73) are of the
z&gt;.
same form
as
the expressions (71) for cr, Z, X ; ra, /* ; 7, n, Moreover, the of condition (59) are equivalent to (72). Hence these equations coefficients are the directioncosines of three fixed directions in
space mutually perpendicular to one another. If lines through the origin of coordinates parallel to these three lines be taken for a new
set of axes, the expressions for #,
reduce to U, V,
7 with reference to these axes ^respectively.* These results may be stated thus
/3,
:
If the general solution of equation (65)
(68)
the curve
"
be
=
STV o
aii
f
whose radii of
is
first
and second curvature
are p
and T
respectively
given by
f\ T)v
wK )
ci
I/
RS/~r&gt;
PS QR
must be remarked that the new axes of coordinates are not necessarily real, so that when it is important to know whether the
It
curves are real
(73).
it
will be advisable to consider the general formulas
this will be given later.
An
shall
example of
We
When
the curve
apply the preceding results to several problems. is plane the torsion is zero, and conversely. For this case equa
tion (65) reduces to
=
ds

of
which the general
ae
integral
is
p
=
where a
is
if J
1
P
=
ae~
&lt;r
i&lt;r
,
an arbitrary constant, and by
(27)
is
the measure of the arc of the
of the
I
spherical indicatrix of the tangent.
This solution
is
form
(08),
with
O
Therefore the coordinates are given by
(75)
x=Ccos&lt;rds,
y=Csir\&lt;rds,
2
=
Hence the coordinates
* This
is
of
any plane curve can be put
as taking
&i
in this form.
the
same thing
ai
=
l,
c*2=
b^
i,
3
oo,
&3=0.
Rechnung auf Geometrie,
t Scheffers,
Anwendung
Leipsic, 1902.
der Differential
und
Integral
Vol.
I,
p. 219.
DETERMINATION OF COORDINATES
We
(65)
29
radii of first
have seon that cylindrical helices are characterized by the property that the und second curvature are in constant ratio. If we put T = pc, equation
bj written
may
^ = _l(l_2c002).
ds
2T
V
Two
2 + 2 cd  1 = 0. particular integrals are the roots of the equation we consider only this case, and put if c is real roots are real and unequal
;
These
(76)
el
= c Vc 2 +
1,
2
=  c + Vc2 +
l,
01&2
=  1.
is
From
(77)
solution of the (69) it follows that the general
above equation
where we have put
(78)
Since
&lt;r
and
 in
(63) are
conjugate imaginary,
if
we
take
then a and 6 must be such that
aeit
_
i
&oe

my _
6.
0j
where
(76)
(7 9 ,
6
denotes the conjugate imaginary of
4
This reduces, in consequence of
to
,
;
n M * = =*
&lt;x&gt;
One
63
solution of this
If these
=
0.
is given by taking values be substituted in (72),
and
for a
4
and
0,
6;
we put a s =
i
GO,
we
get a
kr
where
=
1, 2.
So
that equation (79) becomes && = 0^, where = e0 2 are 61 = 1? 6 2 =  i0i From (77)
i
P
,
= 1, 2. The solutions of this equation Q =  0i, R = e S =  1, so that
,
W==Vc +
2
l
When
a,
/3,
the foregoing values are substituted in (73), and the resulting values of
7
in (61),
we
C
get
(80)
xthe la
Vc2 +
From
a constant ang c
 Ccoslds, J 1
y=

Vc 2 +
1
Jfstafdt,
g
=
Vc 2 +
=
1
v\
ie expressions we find that the tangent to the curve makes the direction of the elements of the cylinder. the zaxis
is
And
the crosssection of the cylinder
Xi
defined by
y\
=
fcos t dsi,
=
J
sin
t
dsi,
where
Si
denotes the arc of this section measured from a point of
it.
If pi
!)
denotes
the radius of curvature of the right section,
we
find that pc 2
=
2
pi(c
+
30
CURVES IN SPACE
EXAMPLES
1.
Find the coordinates
of the cylindrical helix
whose
intrinsic equations are
p
=
T
=
S.
2.
Show
upon a cylinder whose crosssection
3.
that the helix whose intrinsic equations are p is a catenary.
=
T
=
2
(s
+ 4)/V2
lies
p
=
as, r
Establish the following properties for the curve with the intrinsic equations = 6s, where a and b are constants
:
to (a) the Cartesian coordinates are reducible where J_, B, h are functions of a and 6
;
x=Ae ht cos, y = Aeht sm
t,
z
Behi
,
(6)
the curve lies
upon a
circular cone
whose axis coincides with the
zaxis
and
cuts the elements of the cone under constant angle.
16.
Moving
trihedral.
In
11
we took
for fixed axes of refer
ence the tangent, principal normal, and binormal to a curve at a of it, and expressed the coordinates of any other point of Q point the curve with respect to these axes as power series in the arc s is any point of the of the curve between the two points. Since
M
M
curve, there is a set of
such axes for each of
its points.
Hence,
instead of considering only the points whose locus is the curve, we may look upon the moving point as the intersection of three mutually perpendicular lines which move along with the point,
the whole figure rotating so that in each position the lines coin cide with the tangent, principal normal, and binormal at the point.
We shall refer to such a configuration as the moving trihedral. In the solution of certain problems it is of advantage to refer the curve to this moving trihedral as axes. proceed to the con
We
sideration of this idea.
With
cosines of
reference to the trihedral at a point Jf, the directionthe tangent, principal normal, and binormal at
M
have the values
a=l,
As
/3
= 7 = 0;
I
=
0,
m=l,
n
=
0;
X
=p=
0,
i/
= l.
functions with
the trihedral begins to move, the rates of change of these s are found from the Frenet formulas (50) to
have the values
da
ds
_ = =
ft o
d{$
_ =
1
,
dy
ds
_
ft o
dl _ j
1
5
dm _
ds
~ u,
ds
1
j
p
ds
dfJL
p
dn
ds
d\
r~
=
n
"i
r
ds
~T~
1 =~
dv
ds
7~
=
ft U
*
ds
r
MOVING TRIHEDRAL
Let
f,
77 ,
31
f,
f
f denote coordinates referring to the axes at Jf, and those with reference to the axes at Jf , and let JfJf = As
77,
(see fig. 4).
Since the rate of change of a
is
zero
and a
=1
at Jf,
the cosine of the angle between the  and f axes is 1 to within terms of higher than the first order in As. Likewise the cosine of
the angle between the f and Tj axes cosines of the angles between all
is
As/p.
We
calculate the
the axes, and the results tabulated as follows:
f
may be
v
S
As
(81)
.,
_ As
FIG. 4
Let
at
P be
f,
M are
a point whose coordinates with respect to the trihedral 77, f. Suppose that as Jf describes the given curve (7,
P describes
a path T. It
may happen
that in this motion
P is fixed
P
in the relatively to the moving trihedral, but in general the change will be due not only to the motion of the trihedral position of
P
but also to a motion relative to
In the latter general case, if it. on (7, the coordinates the point on T corresponding to denotes and of relative to the axes at may be written
M
P
M
M
?4Af
Thus A 2 #
moving
Tj+A^, f+A^;
and
f
+A
2 f,
17
+A^, f+A
2 ?.
indicates the variation of a function
relative to the
trihedral,
A^
the variation due to the latter and to
the motion of the trihedral.
are within terms of higher order the coordinates of with respect to the axes at Jf, and with the aid of (81) the equations of the transformation of coordinates with respect
(As, 0, 0)
To
M
to the
two axes are expressible thus
.
:
32
CURVES IN SPACE
These reduce to
As
As
.
o
As
.
.
As
T
As
In the limit as
Jtf
As
H
/a
approaches
M these equations become
ds
ds
c?s
p
ds
p
T
ds
ds
0,
T
80
d0
denotes the absolute rate of change of
thus
T
and
=
that
relative to the trihedral.*
If
t
denotes the distance between
2
P and
2
,
a point P^ (f^
find
rj^ fj),
that
is
^
= (? f)
1
2
2
+(7? 1 7;)
+ (? ?) we
1
by means
of the
formulas (82) that
denote the directioncosines of PP^ with respect to the axes at Jf, then
If a,
5,
&lt;?
express the condition that f 1? 77 L ft as well as f, ?/, f satisfy equations (82), we are brought to the following fundamental relations between the variations of a, 6, c:
,
When we
&a
(83)
~T~
c?s
b da ~ ~7
&b
==
T"
db
~T
a I 1
c
*
ds
p
ds
ds
p
T
$c __ dc ~r == ds ds
"T
b
*
r
If the point
P
remains fixed in space as
M moves
are
along the
zero
curve,
the lefthand
members
of
equations (82)
and
the equations reduce to
(84)
ds
l,
p
=
ds
\p
T/
c?s
T
Moreover, the directioncosines of a line fixed in space satisfy
the equations da or .
(85)
b =/)
db
c?s
=
(a
c?s
(\p
c\ +T)
)i
b dc _ = _.
ds
r
Naples, 1896.
* Cf. Cesaro, Lezioni di Geometria Intrinseca, pp. 122128.
MOVING TRIHEDRAL
These are the FrenetSerret formulas,
expected.
as
33
might have been
We
it
shall
show that the
X
1,
the solution of (84).
I,
;
solution of these equations carries with Suppose we have three sets of solutions
7, n, v,
0,
of (85), a,
(86)
& m,
0,
/JL ;
whose values
1,
for s
0,
=
are
0;
0;
0,
1.
They
are the directioncosines, with respect to the moving trihedral with vertex M, of three fixed directions in space mutually perpen
be a fixed point, and through it Let with the directions just found. Take these lines for coordinate axes and let #, y, z denote the coordinates of with respect to them. If f, ??, f denote the coordinates of
dicular to one another.
lines
draw the
M
with respect to the moving trihedral, then f are the ?;, f with respect to the trihedral with vertex at coordinates of
,
M
and edges parallel to the corresponding edges of the trihedral at M. Consequently we have
(87)
=  (ax f &y + rj=(lx+my + m),
f
7*)&gt;
If these
values be substituted in (84) and we take account of are identically satisfied. (50) (85), we find that the equations for s = 0, it follows from If fo ^o ?o Denote the values of f, ?;, (86) and (87) that they differ only in sign from the initial values
and
of x, y,
z.
Hence
if
we
write, in conformity with (21),
(88)
x
and substitute these values in (87), they become the general solu tion of equations (84). We have seen that the solution of equa
tions (85) reduces to the integration of the Riccati equation (65).
17. Illustrative examples. As an example of the foregoing method we consider which is the locus of a point on the tangent to a twisted curve C at a
the curve
constant distance a from the point of contact. The coordinates of the point MI of the curve with reference to the axes at
are a, 0,
(i)
M
0.
In this case equations (82) reduce to
^l^ds~ ds~
a
P
^ds~
34
Hence
if Si
CURVES IN SPACE
denotes the length of arc of C\ from the point corresponding to
s
=
on C, we have
and the directioncosines
iven by are given b
7
of the tangent to Ci with reference to the
moving axes
a2
is
+
p
2
Va2 4 p2
Hence the tangent
point of C.
to Ci
parallel to the osculating plane at the corresponding
By means
ds
of (83)
we
find
p
Sa:\
d
/
\_
+
p
2
a
2 p Va +
ds y V(i 2
p
2
(
2
+
P
2
)
P
Va 2 +
(ii),
p
2
Proceeding in like manner with 0i and
d(*i
71,
and making use
5/3i
of
we have
& 2 pp
(a
2
d
22
)
_ ~
(a
2
2
ftp
p
22
)
p
+
p
a2
+
p
2
5Si
+
p
a2
+ p2
871
_
ap
these expressions and (21 of the first curvature of C\
From
}
we
obtain the following expression for the square
:
app/
Pi
2
a2
+ p 2 \a 2 +
II
t
The
directioncosines of the principal normal of C\ are
5/?i
571
By means
of (40)
:
we
derive the following expressions for the directioncosines of
the binormal
r (a 2
+
p )^
2
r (a 2
+
p )^
2
ft2
&gt;
+ ? Va 2 +
p
2
In order to find the expression for TI, the radius of torsion of Ci, substitute the above values in the equation
we have only
to
_ ~
S\i
_
p
/d\i
\
_ MI\
P
5i
Va2 + p 2
ds
We
leave this calculation to the reader and proceed to an application of the
preceding results.
inquire whether there is a curve necessary and sufficient condition is that
We
C
I/pi,
such that Ci is a straight line. The be zero (Ex. 3, p. 15). From (iii) it
follows that
we must have
ILLUSTRATIVE EXAMPLES
From
the second of these equations
it
35
be plane, and from
follows that
C must
the former
we
get,
by integration,
log (a*
+ ,*) =
+
,
where
take c
c is
=
a constant of integration. If the point s 2 log a , this equation reduces to
=
be chosen so that we
may
P
If 6
=
a
\e 
1.
denotes the angle which the line C\ makes with the axis,
tan 6
we
have, from
(i),
=
8rj
a ==
e
1

1
Differentiating this equation with respect to
s,
we can put
the result in the form
dd__
ds
1
p
consequently
(89)
When
in the
these values are substituted in equations (75),
we
obtain the coordinates of
C
form
x
=
=
i
\1
e
as,
y
= ae
=
a sin
0.
or, in
terms of
0,
(90)
x
a log tan 
+
cos 6
,
y
The curve, with
these equations, is called the tractrix. As just seen, it possesses the property that there is associated with it a straight line such that the segments of the tangents between the points of tangency and points of intersection with the given line are of constant length.
Theorem. The orthogonal trajectories of the osculating plane of a twisted curve can
be
found by quadratures.
plane are (,
reference to the moving axes the coordinates of a point in the osculating The necessary and sufficient condition that this point describe 77, 0). an orthogonal trajectory of the osculating plane as moves along the given curve
With
M
is
that
and
ds ds the equations
in (82) be zero.
Hence we have
for the determination of
and
77
?*_+
da
^+
da,
=
0,
where a
is
given
by
(89).
Eliminating
d?t]
we have
"
^+
=

Hence 77 can be found by quadratures as a function of and then is given directly.
&lt;r,
and consequently of
S,
36
CURVES IK SPACE
Problem. Find a necessary and sufficient condition that a curve
If
,
lie
upon a
sphere.
f denote the coordinates of the center, and R the radius of the sphere, we have 2 f if* + f 2 = R 2 Since the center is fixed, the derivatives of , 17, f are given by (84). Consequently, when we differentiate the above equation, the result = 0, which shows that the normal plane to the curve ing equation reduces to
??,
.
at each point passes through the center of the sphere. If this equation be differen tiated, we get 77 = p hence the center of the sphere is on the polar line for each point. Another differentiation gives, together with the preceding, the following
;
coordinates of the center of the sphere
:
When
(92)
the last of these equations
is
differentiated
we
obtain the desired condition

f (rp
Y
=
0.
Conversely,
lies
when
this condition is satisfied, the point with the coordinates (91)
is
fixed in space
and at constant distance from points of the curve. A curve which upon a sphere is called a spherical curve. Hence equation (92) is a necessary
sufficient condition that
and
a curve be spherical.
EXAMPLES
1.
Show
2.
Let C be a plane curve and Ci an orthogonal trajectory of the normals to C. that the segments of these normals between C and Ci are of the same length. Let
in
C and Ci be two curves in the same plane, and say that the points corre which the curves are met by a line through a fixed point P. Show that if the tangents at corresponding points are parallel, the two curves are similar and P is the center of similitude.
spond
of the point of projection of a fixed point upon the tangent to called the pedal curve of C with respect to P. Show that if r is the makes on (7, and 6 the angle which the line to a point distance from with the tangent to C at M, the arc Si and radius of curvature pi of the pedal
3.
The locus
P
a curve
C
is
P
M
PM
curve are given by
where
s
and
p are the arc
and the radius of curvature of C.
4. Find the intrinsic and parametric equations of a plane curve which is such that the segment on any tangent between the point of contact and the projection of a fixed point is of constant length.
5.
Find the
all
angle
6.
intrinsic equation of the plane curve which meets under constant the lines passing through a fixed point.
is
The plane curve which
of the
such that the locus of the midpoint of the seg
.
ment
normal between a point of the curve and the center of curvature is 2 2 2 line is the cycloid whose intrinsic equation is p f s = a a straight 7. Investigate the curve which is the locus of the point on the principal normal
of a given curve
and at constant distance from the
latter.
OSCULATING SPHERE
18. Osculating sphere.
37
to its
moving
trihedral.
The point whose
Consider any curve whatever referred coordinates have the
values (91) lies on the normal to the osculating plane at the center of curvature, that is, on the polar line. Consequently
the
moving sphere whose center
is
is
at this point,
and whose
radius
cuts the osculating plane in the osculating This sphere is called the osculating sphere to the curve at circle. the point. shall derive the property of this sphere which
f,
Vp
2
r //
2
2
We
accounts for
its
name.
the tangent to a curve at a point is tangent likewise to a sphere at this point, the center of the sphere lies in the normal
When
M
denotes its radius and the curve is plane to the curve at M. If referred to the trihedral at M, the coordinates of the center C of the
2 Let P(x, y, z) sphere are of the form (0, y v z t ) and yl + z* = ^ be a point of the curve near M, and Q the point in which the line CP cuts the sphere. If PQ be denoted by 8, we have, from (53),
.
R
6/r
which reduces to
Hence
?/ 1
8 is of the
second order, in comparison with
JMTP,
unless
is is
=/3, that is, unless the center is on the polar line; then it of the third order unless z l = p r, in which case the sphere the osculating sphere. Hence we have the theorem
:
The osculating sphere to a curve at a point has contact with the curve of the third order ; oilier spheres with their centers on the
polar line, and tangent to the curve, have contact with the curve of the second order ; all other spheres tangent to the curve at a point
have contact of the first order.
The
(93)
radius of the osculating sphere
JS*
is
given by
=,!
+ TV,
in space, are
and the coordinates of the center, referred to fixed axes
(94)
xl = x
+ pi
f
p T\,
y^
= y + pm
p
rfji,
zl
= z + pn
p rv.
88
CURVES IN SPACE
Hence when p
is
and the osculating
circle coincide.
constant the centers of the osculating sphere Then the radius of the sphere is
necessarily constant. Conversely, it follows from the equation (93) that a necessary and sufficient condition that be con IP
E
stant
[P,
is
that
is
is,
either the curvature
is
constant, or the curve
spherical.
If equations (94) be differentiated
with respect to
s,
we get
(96) #1
=
From
sions
it is
these expres seen that
of
the
center
the
is
osculating sphere
fixed
only
in
case
of spherical curves. Also, the tangent to
the locus of the cen
ter
is
parallel to the
binormal.
Combin
with
ing this result
FIG. 5
a previous one, we have the theorem:
is
The polar
line
for a point on a curve
tangent
to the locus
of the
center of the osculating sphere to the curve at the corresponding point.
represented in fig. 5, in which the curve is the are the correspond locus of the points M\ the points (7, C^ C2 are normal ing centers of curvature the planes MCN, M^C^N^ and the are the polar lines the lines CP, C^P^ to the curve
This result
is
,
;
;
;
points P, Pj,
P
2
,
are the centers of the osculating spheres.
BEETEAND CUKVES
19.
39
:
Bertrand curves.
the
To determine
the
Bertrand proposed the following problem curves whose principal normals are the principal
normals of another curve.
moving
trihedral.
We
generate a curve C^ whose principal normal coincides with the ?;axis of the moving remains on the moving ?/axis, we trihedral. Since the point 1
condition that the point
l(
M
In solving this problem we make use of must find the necessary and sufficient
=
0,
TJ
= k, ? = 0)
M
have d%
this
= d% = 0. And since M tends to move at axis, Brj = 0. Now equations (82) reduce to
l
right angles to
(96)
}Ll, ds p
the second
ds
is
vO,
&gt;
5*. ds r
Moreover,
if co
From
we
see that k
a constant.
denotes
makes with the tangent at M, the angle which the tangent at l first and third of these equations, we have, from the
tan
or
sin
co
M
co
8? = ^r =
Sf
cos
co
kp T (k p)
sin
co
(97)
k
have seen ( 11) that according as r is positive or negative, near the osculating plane to a curve at a point cuts the below or above the osculating plane at M. From polar line for
We
M
M
M
these considerations
fourth, or
first
&lt;
it
follows that
when
r
&gt;
&gt;
0, co is in
&lt; &lt;
the third,
&lt;
and when r
accordingly.
k ^, p, or k quadrants according as k co is in the second, first, or fourth quadrant, 0,
;
It is readily
(97).
is
found that these results are consistent
with equation
By means
of (97) it
found from (96) that
the negative sign being taken so that the lefthand member may be positive. Thus far we have expressed only the condition that the locus
of
orthogonally, but not that this axis For this we shall be the principal normal to the curve Cl also. consider the moving trihedral for Cl and let a x b^ c^ denote the
M^ cut the moving
T^axis
,
40
CUKVES IN SPACE
as
directioncosines with respect to it of a fixed direction in space, M^D in fig. 6. They satisfy equations similar to (85), namely
l
(99 )
M
If a,
6, c
are the directioncosines of the
same
direction, with respect to the
&&gt;
mov
ing trihedral at M, we must have a l a cos f c sin &), b l = 6, ^ = a sin eo + cos for all possible cases,
&lt;?
a&gt;,
as
enumerated above.
When
these values are sub
stituted in the above equations, of (98),
sin
tw
_l_
we
get,
by means
_
p
P
sin
I
T
G) &)
sm
sin &)1
\d
\
cos
Tsin
I
I
ft)
a sin
&)
(
&))
=
_
as
c
0,
cos COS
PI
ft)
&lt;w
to
sin
PI
cos
TI
&)1
I
~
[T
*/&gt;
TI J
L
~ ,
*
,
u,
J
dco
~
&)
/3 [sin
cos
&)
k
.
6 h
,
(&lt;?
sm
.
&)
T
r t r sin
+
a cos
&)) =
x
0.
&)J
as
Since these equations must be true for every fixed line, the cients of a, 6, c in each of these equations must be zero. resulting equations of condition reduce to
&)
coeffi
The
= const.,
&)
(100)
sm
cos
&)
1
sm
;;
&lt;w
=
~
"
Since
the
&)
is
first
a constant, equation (97) is a linear relation between and second curvatures of the curve C. And the last of
for the curve Cr equations (100) shows that a similar relation holds a curve C whose first and second curvatures Conversely, given
satisfy the relation
(ioi)
+
p
4,
7
=
c&gt;
where
B,
C
are constants different from zero
;
if
we
take
k
=A
,
COt
ft)
=
B
;&gt;
TANGENT SURFACE OF A CURVE
and
for p l
isfied
41
and r l the values given by (100), equations (99) are sat identically, and the point (0, k, 0) on the principal normal
gather these results generates the curve Cv conjugate to C. about the curves of Bertrand into the following theorem:
necessary and sufficient condition that the principal normals one curve be the principal normals of a second is that a linear of relation exist between the first and second curvatures; the distance
We
A
between corresponding points of the two curves is constant, the oscu lating planes at these points cut under constant angle, and the torsions
of the two curves have the same sign.
We
consider, finally, several particular cases,
which we have
excluded in the consideration of equation (101).
When C =
the curve
that
is,
and
A=Q,
its
the ratio of p and T
is
constant.
Hence
0,
is
a helix and
conjugate
is
at infinity.
When A =
the curve has constant torsion, the conjugate curve coincides with the original. When A = C = 0, k is indeterminate
;
when
hence plane curves admit of an infinity of conjugates, they are the curves parallel to the given curve. The only other curve
which has more than one conjugate is a circular helix, for since p and T are constant, A/C can be given any value whatever both the given helix and the circular helices conjugate to it are traced on circular cylinders with the same axis.
;
20.
Tangent surface of a curve.
For the further discussion of
the properties of curves it is necessary to introduce certain curves and surfaces which can be associated with them. However, in con
sidering these surfaces
we
limit our discussion to those properties
which have to do with the associated curves, and leave other con
siderations to their proper places in later chapters. The totality of all the points on the tangents to a twisted curve
C
constitute the tangent surface of the curve. As thus defined, the sur face consists of an infinity of straight lines, which are called the
on this surface lies on one generators of the surface. Any point of these lines, and is determined by this line and the distance t from
P
P to
arc
,
fig. 7.
the point where the line touches the curve, as is shown in If the coordinates x, y, z of are expressed in terms of the
M
M
the coordinates of
P are
given by
(102)
f
42
CURVES IN SPACE
s.
where the accents denote
differentiation with respect to the equations of the curve have the general form
When
the coordinates of
(103)
P
can be expressed thus
:
=./+/,
v
where
=
(104)
?=/,() +./,
f /,()
From
is
this it is seen that v is equal to the distance
MP only when s
the parameter.
As given by equations (102) or (103), the coordinates of a point on the tangent surface are functions of two parameters. A rela tion between these parameters, such as
f(s, t)
= 0,
upon
this
defines a curve which lies
the surface.
FIG. 7
For,
when
t
equation
is
solved for
in terms
sion
of s and the resulting expres substituted in (102), the coordinates f, ?;, f are functions of a single parameter, and consequently the
is
locus of the point (f, 77, f) is a curve (1). By definition, the element of arc of this curve
da2
= di;
2
f
drf +
2
c?f
.
This
is
is given by by means of (102) and expressible
(41) in the
form
z
d&lt;r
(105)
=
l
+
2
ds
2
+
2dsdt
+
dt\
where t is supposed to be the expression in s obtained from (104), and p is the radius of curvature of the curve (7, of which the sur face is the tangent surface. This result is true whatever be the
relation (104). Hence equation (105) gives the element of length of any curve on the surface, and do is called the linear element of
the surface.
in equations (102) has a positive or negative value, the point lies on the portion of the tangent drawn in the
According as
t
TANGENT SUBFACE OF A CUKVE
43
in the opposite direction. It positive direction from the curve or is now our purpose to get an idea of the form of the surface in the
neighborhood of the curve. In consequence of (53) equations (102) can be written
1
\
L
^r+..u,
1
6 pr
The
for f
plane f
s
it is
at which
= = 0,
cuts the surface in a curve F.
is
also a point of F.
From
The point Q of (7, the above expression
M
M
t
seen that for points of
s
F near
only in
s
Q
the parameters
and
t
differ
sign.
Hence, neglecting powers of
and
of higher orders, the equations of the neighborhood of J/ are
F
in
f=0
By
we
,
,=., r=_ 2/o
t
&lt;
3 pr
eliminating
from the
last
two equa
find that in the neighborhood of tions, the curve F has the form of a semiQ
M
cubical parabola with the T^axis, that is the principal normal to (7, for cuspidal
tangent. Since any point of the curve C can be taken for Jf we have the theorem
,
:
The tangent surface of a curve
consists of two sheets, corresponding
which are tangent to respectively to positive and negative values of t, one another along the curve, and thus form a sharp edge.
On
this
account the curve
is
called the edge of regression of the
surface.
An
idea of the form of the surface
may be had from
fig.
8.
21. Involutes and evolutes of a curve. When the tangents of a curve C are normal to a curve Cv the latter is called an involute of
(7,
and C
is
called an evolute of
Cr As
of a twisted curve lie
upon
its
thus defined, the involutes of a tangent surface, and those
44
plane curve in
CURVES IN SPACE
its plane. The latter is only a particular case of the former, so that the problem of finding the involutes of a curve is that of finding the curves upon the tangent surface which cut the
generators orthogonally. write the equations of the tangent surface in the form
We
Assuming
that
s is
to the determination of a relation
the parameter of the curve, the problem reduces between t and s such that
ds c of (50) this reduces to dt 0, so that t s, where c is an arbitrary constant. Hence the coordinates x^ y v z l of an involute are expressible in the form
(106)
By means
+
=
2^= x
+ a(c
s),
#!
=#+
(&lt;?),
zx
=z+
;
?(&lt;?
s).
Corresponding to each value of c there is an involute consequently a curve has an infinity of involutes. If two involutes correspond
to values
c^
and
c 2 of
c,
curves
is
of length c l
c2 .
the segment of each tangent between the Hence the involutes are said to form a
system of parallel curves on the
tangent surface.
When s is known the involutes
by equations the complete de termination of the involutes of a
(106).
are given directly
Hence
given curve requires one quad rature at most.
FIG. 9
its
From
the definition of
t
and
above value, an involute can
be generated mechanically in the following manner, as represented in fig. 9. Take a string of length c and bring it into coincidence
call the other with the curve, with one end at the point s = end A. If the former point be fixed and the string be unwound trace out gradually from the curve beginning at A, this point will
;
an involute on the tangent surface.
By
differentiating equations (106),
we
j
ds,
get
dz l
,
dx l
7
=
I
(c
s}
j ds,
,
dy l
= m (c
s)
n (c 
s)
,
ds.
INVOLUTES AND EVOLUTES
Hence the tangent
45
to an involute is parallel to the principal nor
mal of the curve at the corresponding point, and consequently the tangents at these points are perpendicular to one another.
As an example of the foregoing theory, cular helix, whose equations are
x
we determine
the involutes of the cir
=
a cos te,
y
=
a sin M,
z
=
au cot 0,
where a
is
the curve
makes with
s
the radius of the cylinder and 6 the constant angle which the tangent to axis of the cylinder. Now
a cosec 6

u,
a,
/3,
7
=
 sin u, cos M, cot :
cosec
Hence the equations
Xi
of the involutes are
c sin 0)sin M,
=
a cos u
+
(au
yi
it
=
a sin u
(aw
c sin 0) cos u,
zi
=
c cos
0.
From
the last of these equations
follows that the involutes are plane curves
whose planes are normal to the axis of the cylinder, and from the expressions for x\ and yi it is seen that these curves are the involutes of the circular sections of
the cylinder.
We
proceed to the inverse problem
C, to find its evolutes.
:
Given a curve
normals to
The problem reduces to the determination of a succession of C which are tangent to a curve G If Q be the point on (7, it lies in the normal plane to C at on C corresponding to
.
M
M
If,
and consequently
its
coordinates are of the form
where p and q are the distances from Q to the binomial and prin cipal normal respectively. These quantities p and q mi^t be such that the line is Q tangent to the locus of Jf at tiis point, that
M
MM
is,
we must have
,,
where
/c denotes a factor of proportionality. values are substituted in these equations, we
"When
the above
et
P
and two other equations obtained by replacing a, Z, X by /3, m, ft and 7, n, v. Hence the expressions in parentheses vanish. From
46
the
first it
CURVES IN SPACE
follows that
p
the polar line of , written
ds
C
at
M.
lies on equal to /o; consequently Q The other equations of condition can be
is
M
dp q J: + 1 + p
r
da *
a,
,
ds
_p+
T
0&gt;
Eliminating
/c,
we get
p
For the sake of convenience we put
integration
&lt;o
=
I

&gt;
and obtain by
P
= tan
(o&gt;
+
c),
where c is the constant of integration. As c is arbitrary, there is an infinity of evolutes of the curve C\ they are defined by the following equations, in which c is constant for an evolute but
changes with
it:
xQ =x+lp
+ \p tan(o) + c), y =y + mp + fip Z = z f np + vp tan c).
Q
tan(o&gt;
f c),
Q
f
(o&gt;
From
the definition of q
it
of the angle
which
MM
:
follows that q/p
is
equal to the tangent
to
Q
makes with the principal normal
6
C at M.
Calling this angle 0,
we have
=
&&gt;
+
c.
The foregoing
results give
the following theorem
A
curve
f"
C
(7,
normals
admits of an infinity of evolutes; when each of the which are tangent to one of its evolutes, is turned
to C, these
through the sa^g angle in the corresponding normal plane new normals are tangent to another evolute of C.
In
fig.
5 the locus of the points
o;
E
is
an evolute of the given
curve.
Each system
normals to
C which
are tangent to an evolute
C
constitute a tangen t surface of which C is the edge of regression. Hence the evolves of C are the edges of regression of an infinity
of tangent svff aces? all o f
which pass through
follows that w
is
C..
From
is
tbe definition of
w
it
constant only when the curve
C
we have plane. j n this case we may take w equal to zero. Then when c the evol\te C in the plane of the curve. The other evolutes lie upon the right
MINIMAL CUEVES
47
cylinder formed of the normals to the plane at points of Co, and cut the elements of the cylinder under the constant angle 00 c, and consequently are helices.
Hence we have the theorem
:
The evolutes of a plane curve are the helices traced on the right cylinder whose base is the plane evolute. Conversely, every cylindrical helix is the evolute of an
infinity of
plane curves.
EXAMPLES
1.
Find the coordinates of the center of the osculating sphere of the twisted
cubic.
2. The angle between the radius of the osculating sphere for any curve and the locus of the center of the sphere is equal to the angle between the radius of the
osculating circle and the locus of the center of curvature.
3.
The locus
is
of the center of curvature of a curve is
an evolute only when the
curve
4.
plane.
first
Find the radii of
y
=
a cos 2 it, z
=
construction.
5.
asinw. Show that the curve Find its evolutes.
and second curvature of the curve x = a sin u cos w, is spherical, and give a geometrical
without the use of the moving
Derive the properties of Bertrand curves
(
10)
trihedral.
6.
Find the involutes and evolutes of the twisted cubic.
Determine whether there
Derive the results of
is
7.
a curve whose bmormals are the binormals of a
second curve.
8.
21 by means of the moving trihedral.
Minimal curves. In the preceding discussion we have made exception of the curves, defined by
22.
z =/! (w),
y
=/
a
(u),
z
=/
8
(u),
when
these functions satisfy the condition
As
these imaginary curves are of interest in certain parts of the theory of surfaces, we devote this closing section to their discussion.
The equation
of condition
may
be written in the form
_ J3 f
where
equivalent to the following
(108)
:
Jl
f __ if V2
These equations are
v is a constant or a function of u.
^ ^lz*
:
SI.&lt;l+*&gt;:..
48
CURVES IN SPACE
At most, the common ratio is a function of M, say f(u). And so we disregard additive constants of integration, as they can be removed by a translation of the curve in space, we can replace
if
the above equations by
(1 09)
x
=.f(u)du,
first
y
=
if(
2
n)du,
z
=
call it a.
We
consider
the case
when
v is constant
and
I
If
we change the parameter of the curve by replacing new parameter which we call w, we have, without loss
(110)
f(u) du by a
of generality,
1a x=
2
t
^u
y
= .1+a i^u,
z
= au.
For each value of a these are the equations of an imaginary straight line through the origin. Eliminating #, we find that the
envelope of these lines
origin,
is is
the imaginary cone, with vertex at the
whose equation
(111)
z2 +2/ 2 +z 2 =0.
Every point on the cone is at zero distance from the vertex, and from the equations of the lines it is seen that the distance between any two points on a line is zero. We call these generators of the
cone minimal straight
lines.
Through any point
in space there are
an infinity of them
;
their directioncosines are proportional to
&gt;
where a
vertex
is
is
arbitrary.
The
locus of these lines
is
the cone whose
and whose generators pass through the circle at infinity. For, the equation in homogeneous coordinates of the 2 2 = w2 sphere of unit radius and center at the origin is 3? + y + z
at the point
,
so that the equations of the circle at infinity are
Hence the cone
of u.
If
(111) passes through the circle at infinity.
this function of
it,
We consider now the case where v in equations (109) is a function
we take
u
for a
new
convenience
(112)
call it
equations (109)
may
parameter, and for be written in the form
g
_ll~p
&gt;()&lt;**,
y = i^^F(u)du,
z=
CuF(u)du,
MINIMAL CUEVES
where, as
is
49
seen from (108), F(u) can be any function of u different
third derivative of a function f(u),
from
If
zero.
we replace F(u) by the
thus
F(u)=f
form
"(u],
equations (112) can be integrated by parts and put
in the
uf (u)f(u),
(113)
1y4
Since
F must be
c
l u*+
than
c 2 u f c8 ,
different from zero, f(u) can have any form other where ^, c 2 , c3 are arbitrary constants.
EXAMPLES
that the tangents to a minimal curve are minimal lines, and that a curve whose tangents are minimal lines is minimal.
1. 2.
Show
Show
that the osculating plane of a minimal curve can be written
Q,
+ (Yy)B + (Zz)C =
of this sort
3.
is
where
A + B2 + C = 0. A plane
2
2
(X
x)
A
is
whose equation
called an isotropic plane.
Show
lie
that through each point of a plane two minimal straight lines pass
which
4.
in the latter.
Determine the order of the minimal curves for which the function /in (113)
condition
satisfies the
5.
4/
v
"/
 5/ iv2 =
,
0.
Show that the equations of a minimal v condition 4/ 5/ iv2 = a/ ///3 where a is
"/
curve, for which /in (113) satisfies the
a constant, can be put in the form
.
x
=
8  cos
,
y
8  sin
,
z
=
8i, t.
GENERAL EXAMPLES
1.
Show
that the equations of any plane curve can be put in the form
x=J*cos0/(0)d0,
2.
y
J
0.
sin
0/(0) d0,
and determine the geometrical significance of
Prove that the necessary and sufficient condition that the parameter u in the in Ex. 1 is equations x =fi(u), y =f2 (u) have the significance of
3. Prove that the general projective transformation transforms an osculating plane of a curve into an osculating plane of the transform.
4.
The
principal normal to a curve
is
is
normal
to the locus of the centers of
curvature at the points where p
a
maximum
or
minimum.
50
.
CURVES IN SPACE
A
5 certain plane curve possesses the property that if C be its center of curva ture for a point P, Q the projection of on the xaxis, and T the point where the meets this axis, the area of the triangle is constant. Find the tangent at equations of the curve in terms of the angle which the tangent forms with the xaxis.
P
P
CQT
6. The binormal at a point Mot a curve is the limiting position of the as and approaches M. perpendicular to the tangents at
common
M
M
,
N
7. The tangents to the spherical indicatrices of the tangent and binormal of a twisted curve at corresponding points are parallel.
8. Any curve upon the unit sphere serves for the spherical indicatrix of the binormal of a curve of constant torsion. Find the coordinates of the curve.
9.
The equations r Idk  kdl x a J #2 + 2 + 12
I

i
y
I
a
r hdl
I
fc2
Idh
z
J hZ
d
r kdh
I
~
hdk
+
+
12
J h2
+
where a is constant and h, k, whose radius of torsion is a.
10. If, in Ex. 9,
are functions of a single parameter, define a curve
we have
k
=
sm/i0
+ %sinX0
is
I
=
2
\/
cos
2
commensurable, the integrands are expressible as linear homogeneous functions of sines and cosines of multiples of 0,
/x
where X and
are constants whose ratio
and consequently the curve
is
algebraic.
t.
11. Equations (1) define a family of circles, if a, &, r are functions of a parameter Show that the determination of their orthogonal trajectories requires the solution
of the Riccati equation,
*!
dt
= l*?,__L*
r dt
where 0=tanw/2.
12.
8rdr i"),
(
Find the vector representing the rate of change of the acceleration of a
point.
moving
13.
When
a curve
is
of the perpendicular
upon the osculating plane
spherical, the center of curvature for the point is the foot at the point from the center of the
sphere.
14. The radii of first and second curvature of a curve which lies upon a sphere 2r = 0, and cuts the meridians under constant angle are in the relation 1 + ar + b are constants. where a and
fy&gt;
An epitrochoidal curve is generated by a point in the plane of a circle which without slipping, on another circle, whose plane meets the plane of the first rolls, circle under constant angle. Find its equations and show that it is a spherical curve.
15. 16. If
two curves are
in a onetoone correspondence with the tangents at
are parallel corresponding points parallel, the principal normals at these points and likewise the binormals two curves so related are said to be deducible from one another by a transformation of Combescure.
;
17. If two curves are in a onetoone correspondence and the osculating planes at corresponding points are parallel, either curve can be obtained from the other by a transformation of Combescure.
GENERAL EXAMPLES
18.
51
Show
[x""
E2
T2p4
2
that the radius of the osculating sphere of a curve is given by + /z + z ///2 ] r 2 where the prime denotes differentiation with
y"
,
respect to the arc.
its
At corresponding points of a twisted curve and the locus of the center of osculating sphere the principal normals are parallel, and the tangent to one curve is parallel to the binorinal to the other also the product? of the radii of torsion of the two curves is equal to the product of the radii of first curvature,
19.
;
or to within the sign, according as the positive directions of the principal normals are the same or different.
20. Determine the twisted curves which are such that the centers of the spheres osculating the curve of centers of the osculating spheres of the given curve are points of the latter.
21.
Show
that the binormals to a curve do not constitute the tangent surface
of another curve.
22. Determine the directions of the principal
of a given curve.
normal and binormal
to
an involute
23.
Show
x
that the equations
(u) sin
=
a
C&lt;f&gt;
u du,
y
=
^
2
a
)
\
$ (u) cos u du,
and
\f/
z
a f 4&gt;(u)\l/(u)du,
where
(u)
(1 4
^2
4
^
/2
) (1 4
*
(u) is
any function whatever, define a
curve of constant curvature.
24. Prove that
when ^ (u) =
tan w, in example 23, the curve
is
algebraic.
25. Prove that in order that the principal normals of a curve be the binor
mals of another, the relation a I
stants.
h
=
)

must hold, where a and
of
6 are
con
Show
that such curves are defined
by equations
(1
__
example 23 when
.
=
26. Let \i,
/ii,
*i
(1 4.
^2
__
^/2)3
2
i//
__
^2)3(^" _j_
^,\2
(1 4.
)^(l 4
2
1//
4 1// )^
2
be the coordinates of a point on the unit sphere expressed as
&lt;TI
functions of the arc
of the curve.
Show
that the equations
/
x
=
ek
I
\idffi
k cot w
k cot w k cot w
(MI^I
y
z
= =
ek
j
I
mdai
vidai
\
(v\\{
v{\\]
d&lt;?i,
ek

(\i/4
where k and w are constant, e = 1, and the primes indicate differentiation with respect to oi, define a Bertrand curve for which p and T satisfy the relation (97) show also that X 1? /t 1? v\ are the directioncosines of the binormal to the conjugate
;
curve.
CHAPTER
II
CURVILINEAR COORDINATES ON A SURFACE* ENVELOPES
In the preceding chapter seen that the coordinates of a point on the tangent surface of a curve are expressible in the form
23. Parametric equations of a surface.
(1)
we have
x
where
f
l
(u),
?=/(*),
are the equations of the curve, and v is proportional to the distance between the points (f 77, f ), (x, y, z) on the same generator. Since
,
the coordinates of the surface are expressed by (1) as functions of
two independent parameters
equations of the surface be written
w, v,
the
may
Consider also a sphere of radius
a whose center
(fig.
is
at the origin
10).
If v denotes the angle,
measured
in the positive sense,
which the plane through the zaxis FIG. 10 of the sphere makes and a point with the #zplane, and u denotes the angle between the radius OM and the positive zaxis, the coordinates of may be written
M
M
(3)
x
= a sin u cos v,
y
= a sin u sin v,
z
= a cos u.
the
Here, again, the coordinates of any point on the sphere are ex
pressible as functions of
two parameter^, and the equations of
sphere are of the
form
(2)*.
is
* Notice that in this case /^
a function of u alone.
PARAMETRIC EQUATIONS OF A SURFACE
,
53
In the two preceding cases the functions fv /2 /3 have par consider the general case where /1? /2 /3 are ticular forms. any functions of two independent parameters w, v, analytic for all
We
v,
,
values of u and
or at least for values within a certain domain.
The
locus of the point whose coordinates are given by (2) for all values of u and v in the domain is called a surface. And equa
tions (2) are called parametric equations of the surface. It is to be understood that one or more of the functions
/ may
be
involve a single parameter. For instance, any cylinder defined by equations of the form
may
x =fi
If
M
=F
l
y
=/ M
2
z
=/a u
(
^
v )
u and v in (2) by independent functions other parameters u v v v thus
replace
(4)
we
of
two
u
(u v v,),
v
=F
z
(u l9 vj,
the resulting equations
(5)
may
y
be written
x
= fa (u^
VJ,
= fa K,
vj,
z
= fa (u
t,
vj.
If particular values of
ing values of
of
#, y, z
u and
v l be substituted in (4) and the result v be substituted in (2), we obtain the values
^
and
given by
ticular values.
face,
and t^ have been given the par (5), when u^ Hence equations (2) and (5) define the same sur
are of such a form that fa, fa, fa s. Hence the satisfy the general conditions imposed upon the equations of a surface may be expressed in parametric form in
provided that
F
l
and
F
2
F
the
number
of
ways
y,
Suppose the
terms of x and
solutions.
first
two arbitrary functions. two of equations (2) solved for u and v in and let u = Ft (x, y), v = F2 (x, y) be a set of
of the generality of
When
(5)
these
equations are
taken as equations
(4),
equations
become
x
= x,
y
= y,
=/(*,
z
=f(x,
y},
which may be replaced by the single
(6)
relation,
2
y).
first
If there is
only one set of solutions of the
two
of equations
(2),
equation (6) defines the surface as completely as (2). If, however, there are n sets of solutions, the surface would be defined by n
equations, z
=f (x^
t
y).
54
It
CURVILINEAR COORDINATES ON A SURFACE
may
be said that equation (6)
is
obtained from equations
(2)
by eliminating u and v. This is a particular form of elimination, the more general giving an implicit relation between x, y, z, as
(7)
F(x,y,z)=0.
If
we have
form
of the
a locus of points whose coordinates satisfy a relation For, if we take (6), it is a surface in the above sense.
v,
/
x and y equal to any analytic functions of u and and substitute in (6), we obtain z =/8 (w, v). 2
,
namely
f^
and
In like manner equation
(7)
may
be solved for
z,
and one or more
equations of the form (6) obtained, unless z does not appear in (7). In the latter case there is a relation between x and y alone, so that
the surface
and
its
a cylinder whose elements are parallel to the zaxis, parametric equations are of the form
is
x =/i
W
y
=/
2
M
2
=/ (w,
3
v).
Hence
(6),
or
a surface can be denned analytically by equations (2), Of these forms the last is the oldest. It was used (7).
exclusively until the time of Monge, who proposed the form (6); the latter has the advantage that many of the equations, which
define properties of the surface, are simpler in form than when equation (7) is used. The parametric method of definition is due
to Gauss.
In
It
many
will
respects
it
is
methods.
treatment.
be
used
almost
superior to both of the other entirely in the following
24. Parametric curves.
is
When
the parameter
u
in equations (2)
put equal to a constant, the resulting equations define a curve on the surface for which v is the parameter. If we let u vary continu
ously, we get a continuous array of curves whose totality consti tutes the surface. Hence a surface may be considered as generated
by the motion of a curve. Thus the tangent surface of a curve
is
described by the tangent as the point of contact moves along the curve and a sphere results from the revolution of a circle about
;
a diameter.
have just seen that upon a surface (2) there of curves whose equations are given by equations
constant, each constant value of
We
lie
(2),
an
infinity
is
when u
u determining a curve.
We
call
them the curves u
=
const,
on the surface.
In a similar way,
PARAMETRIC CURVES
there
55
The curves of is an infinite family of curves v = const.* two families are called the parametric curves for the given these equations of the surface, and u and v are the curvilinear coordinates
say that the positive direction upon the surface.f of a parametric curve is that in which the parameter increases. If we replace v in equations (2) by a function of w, say
of a point
(8)
We
v
#, y, z are
=
&lt;t&gt;(u),
functions of a single parameter w, and consequently the locus of the point (#, y, z) is a curve. Hence equation (8) defines a curve on the surface (2). For example,
the coordinates
the equation v
= au defines x = a cos w,
(8) is
a helix on the cylinder
y
o&gt;
sin u,
z
= v.
Frequently equation
(9)
written in the implicit form,
v)
F(u,
= 0.
of this form.
Conversely, any curve upon the surface is defined by an equation For, if t be the parameter of the curve, both u and v in
equations
(2)
are functions of
t\
thus w
=^
1
(Q, v
=
(j&gt;
z (t).
Elimi
t between these equations, we get a relation such as (9). return to the consideration of the change of parameters, defined by equations (4). To a pair of values of u^ and v l there
nating
We
correspond unique values of u and v. On the contrary, it may happen that another pair of values of u^ and v l give the same values of u and v. But the values of x, y, z given by (5) will be
the same in both cases
;
this follows
from the manner in which
these equations were derived. On this account when equations (4) are solved for u^ and v l in terms of u and v, and there is more
than one set of solutions,
be used.
(10)
we must
specify which solution will
We
write the solution
u^
=
&lt;$&gt;!
(w, v),
v^
=
4&gt;
2
(u, v).
In terms of the original parameters, the parametric lines u^= const. and v l = const, have the equations,
*
On
the sphere defined
by equations
(3)
the curves v
const, are meridians
and
u
t
const, parallels.
When
a plane
two families
is referred to rectangular coordinates, the parametric lines are the of straight lines parallel to the coordinate axes.
56
CURVILINEAR COORDINATES ON A SURFACE
b
where a and
denote constants.
Unless u or
v is
absent from
either of these equations the curves are necessarily distinct from the parametric curves u const, and v const. Suppose, now, that
=
=
v does not appear in
and vice
versa.
then u^ is constant when u is constant, Consequently a curve u^ = const, is a member of
^j
the family of curves
parameters
is
Hence, when a transformation of made by means of equations of the form
u
= const.
or
^=(2,,
^(M),
the two systems of parametric curves are the same, the difference being in the value of the parameter which is constant along a curve.
EXAMPLES
1
.
A surface which is
;
straight line orthogonally
right conoid
its (v)
the locus of a family of straight lines, which meet another and are arranged according to a given law, is called a
that
2.
when
=
equations are of the form x = u cos v, y = u sin u, z a cot v + b the conoid is a hyperbolic paraboloid.
is
=
&lt;j&gt;
(v).
Show
passes through the ellipse x
3.
Find the equations of the right conoid whose axis z2 V 2 "
the axis of
z,
and which
a,
\
1.
When
a sphere of radius a
is
defined
by
(3), find
the relation between
f
u and
a4
.
v along the curve of intersection of the sphere and the surface x4 Show that the curves of intersection are four great circles.
4.
y*
+
z4
=
Upon
the surface x
v w2 +
J
cos
t&gt;,
y
Ma
4
sin v, z
=
w,
determine the
that two and second
curves whose tangents make with the zaxis the angle tan 1 \/2. of these curves pass through every point, and find their radii of curvature.
Show
first
25.
is
Tangent plane.
A
tangent line to a curve upon a surface
called a tangent line to the surface at the point of contact. It is evident that there are an infinity of tangent lines to a surface at a shall show that all of these lines lie in a plane, which point.
We
is
called the tangent plane to the surface at the point. To this end we consider a curve C upon a surface
y, z)
and
let
M(XJ
be the point at which the tangent
(
is
drawn.
The
equations of the tangent are
4)
fs = t)y _ ?g = ^
dx
ds
dy
ds
dz
ds
TANGENT PLANE
where
f,
77,
57
for their values
f are the coordinates of a point on the line, depending upon the parameter X. If the equation in curvi
linear coordinates of the curve
C
is
v
=
*
&lt;f&gt;(u),
the above equations
may
be written
.
(
^
,
,
\ dx\
j du
dvl ds
=^ \
\du
f
^^
4&gt;
cv/ ds
)r
In order to obtain the
where the prime indicates
equations.
differentiation.
locus of these tangent lines,
we
eliminate
$
arid
X from these
This gives
(U)
=
0,
which evidently is the equation of a plane through the point M. The normal to this plane at the point of contact is called the normal to the surface at the point.
As an example, we
of a curve at
find the equation of the tangent plane to the tangent surface
If the values
any
point.
is
from
(1)
be substituted in equation
(11),
the resulting equation
reducible to
(12)
/i
fi
fi
fi
is
fi
fs
Hence the equation
upon
u.
of the tangent plane
(I,
In consequence of
to the
36)
*
we have
independent of the theorem
:
u,
and depends only
The tangent plane
generator; touches the curve.
it
is the
tangent surface of a curve is the same at all points of a osculating plane of the curve at the point where the generator
When the surface is defined by an equation of the form F(x, y, z) = 0, we to imagine that x, y. z are functions of u and v, and differentiate with respect the latter. This gives
~
Hx du
dy du
dz du
dx dv
dy dv
dz dv
* In references of this sort the
Roman numerals
refer to the chapter.
58
CURVILINEAR COORDINATES ON A SURFACE
of these equations the equation (11) of the tangent plane can be given
By means
the form
(l^+fojO^+tf*)?^. ex cz cy
When
it is
(
the
Monge form
of the equation of a surface, dz
namely
z
=/(x, y\
t
is
used,
customary to put
rdx
14
= P
cz
=
is
9
cy
Consequently the equation of the tangent plane
(15)
(*

x)p
+
(77

y)q
(fz) =
is
0.
In the
(16)
first
chapter we found that a curve
defined
by two equations of the form
F
l
(x,y,z)
=
J
F2 (x,
y, z)
=
0.
Hence a curve is the locus of the points tions of the tangent to the curve are
common
to
two surfaces.
The equa
gX^q y _ Z
dx
^
dy
dz
where
cfcc,
dy, dz satisfy the relations
5*1* +
dx
^dy + ?*& = cz cy
z
0,
?**, +
dx
^dy + ?** = dz cy
$
0.
Consequently the equations of the tangent can be put in the form
(17)
77

y
dx
dz

z
dy
.
dz
dz
dy
(13),
dz
dx
dx
dy
dy
dx
Comparing
is
this result
with
we
the intersection of the tangent planes at the curve.
see that the tangent line to a curve at a point to two surfaces which intersect along
M
M
*"
EXAMPLES
1.
Show
that the volume of the tetrahedron formed
by the coordinate planes and
y
the tangent plane at any point of the surface x
2.
=
w,
=
u,
z
=
a s/uv is constant.
Show
that the
sum
of the squares of the intercepts of the axes
by the tan
gent plane to the surface
z
at
= w 3 sin 3 u,
y
=
M 3 cos 3 v,
z
=
(a
2

it
2 )*,
any point
3.
is
constant.
Given the right conoid for which 0(u) = a sin 2 u. Show that any tangent plane to the surface cuts it in an ellipse, and that if perpendiculars be drawn to the generators from any point the feet of the perpendiculars lie in a plane
ellipse.
ENVELOPES
4.
59
Show
(u)
which
5.
= a Vtan u,
that the tangent planes, at points of a generator, to the right conoid for in parallel lines. meet the plane z
to the curve
Find the equations of the tangent
ax
2
whose equations are
+
by*
+
cz 2
=
1,
6x 2
+
cy
2
+
az 2
=
1.
6.
Find the equations of the tangent
z(x
to the curve
whose equations are
a)
+
z)(x
is
a)
=
a
3
,
z(y
+
z)(y
=
a3
,
and show that the curve
7.
plane.
The distance from a point
M
is
point
M
is
of the second order
when
through
M the distance from M
form
of a surface to the tangent plane at a nearby is of the first order and for other planes
MM
;
ordinarily of the
first
order.
26. Oneparameter families of surfaces.
of the
(18)
Envelopes.
An equation
F(x, y, z,a)
=
Q
value of the parameter
defines an infinity of surfaces, each surface being determined by a Such a system is called a oneparameter a.
For example, the tangent planes to the tangent surface of a twisted curve form such a family. The two surfaces corresponding to values a and a of the param
family of surfaces.
eter
meet in a curve whose equations may be written
&gt;
*
a)
= o.
a
a
As
a approaches
1
a, this
curve approaches a limiting form whose
equations are
(19)
^(
W
,)=0,
is
*(**.)().
The curve thus
a.
defined
called the characteristic of the surface of
a family of these characteristics, parameter and their locus, called the envelope of the family of surfaces, is a a varies
As
we have
surface
whose equation
is
equations (19). This elimination the second of (19) for a, thus:
a
obtained by eliminating a from the two may be accomplished by solving
= $ (#,
y, z),
and substituting in the
first
with the result
60
ENVELOPES
The equation
of the tangent plane to this surface
is
in
For a particular value of a, say a equations (19) define the curve which the surface F(x, y, z, a ) = meets the envelope and from
,
;
the second of (19) it follows that at all points of this curve equa tion (20) of the tangent plane to the envelope reduces to
This, however,
is
F(x,
y, z,
a
)
= 0.
the equation of the tangent plane to the surface If we say that two surfaces with the same tan
gent plane at a
common
point are tangent to one another,
we have
is
:
to
The envelope of a family of surfaces of one parameter each surface along the characteristic of the latter.
tangent
The equations of the characteristic of the surface of parameter a l are
(21)
This characteristic meets the characteristic (19) in the point whose
coordinates satisfy (19) and (21),
tions (19)
or,
what
is
the
same thing, equa
and
z,
F(x,y,
a l )F(x,^z, a)
a l approaches 0, this point of intersection approaches a limiting position whose coordinates satisfy the three equations
(22)
As
F0,
^=0, da
a;,
!"
da
2
0.
If these equations
be solved for
y, z,
we have
*=/.(a).
is
(23)
*
= /,
y=/,(a),
These are parametric equations of a curve, which
edge of regression of the envelope.
called the
DEVELOPABLE SURFACES
The
61
directioncosines of the tangent to the edge of regression

are proportional to
da
^, . da da
If
we imagine
that
x, y, z
in (19)
are replaced by the values (23), and we differentiate these tions with respect to a, we get, in consequence of (22),
equa
dx da
dy da
tfF dy da dy da

dz
da
dz
==
"
*r
da d#
**
,
c?#
da dz da
From
these
we
obtain
follows that the minors of the righthand mem ber are proportional to the directioncosines of the tangent to the
But from
curve
(17)
it
(19).
Hence we have the theorem:
The characteristics of a family of surfaces of one parameter are
tangent
to the
edge of regression.
Rectifying developable. simple ex ample of a family of surfaces of one parameter is afforded by a family of planes of one parameter. Their envelope is called a developable surface ; the full significance of this term will be
27. Developable surfaces.
A
shown
later
(43).
The
characteristics are straight lines
which
are tangent to a curve, the edge of regression. When the edge of regression is a point, the surface is a cone or cylinder, according as the point is at a finite or infinite distance. exclude this
We
case for the present and assume that the coordinates x, y, z of a point on the edge of regression are expressed in terms of the arc s. may write the equation of the plane
We
(24)
(X x)a + (Y,
where
by
5,
c also are
functions of
its
s.
The
characteristics are defined
s,
this equation
and
derivative with respect to
namely
:
(25)
(X x)a +(Y y)V+(Z z)c  ax  by 
cz
= 0.
62
ENVELOPES
Since these equations define the tangent to the curve, they must be equivalent to the equations
Xx _Yy = Zz
x
y
r
z
Hence we must have
(26)
ax
+%+
cz
=
0,
ax
+ Vy + c z = 0.
If the first of these equations
the resulting equation
of (26), to
is
be differentiated with respect to s, reducible, in consequence of the second
if
ax"
+
E
jf
by"
+
,
IT
cz"=
A 0.
From
this
equation and (26)
ab
: :
we
:
find
(z
x"
c
= (y
z"
z
y")
x
z")
:
(x
y"
y
x"}.
Hence by
(
7)
we have
the theorem:
On
the envelope of a oneparameter family of planes the planes
osculate the edge of regression.
We
itself.
leave
it
to
the
sion of the osculating planes of a twisted curve
reader to prove that the edge of regres is the curve
normal to the principal normal to a curve at a point of the curve is called the rectifying develop able of the latter. We shall find the equations of its edge of
The envelope
of the plane
regression.
The equation
(27)
If
of this plane
I
is
(X x) + (Y y) m + (Z  z) n = 0.
curve,
we differentiate this equation with respect to the arc of the and make use of the Frenet formulas (I, 50), we obtain
(28)
(I

+
we
derive the equations of the character
From
istic in
these equations the form
RECTIFYING DEVELOPABLE
t
63
being the parameter of points on the characteristic. In order to find the value of t corresponding to the point where the character istic touches the edge of regression, we combine these equations with the derivative of (28) with respect to s, namely
:
and obtain
(jLJ\t +  s*Q. P VP PT/
the coordinates of the edge of regression of the rectifying developable are
(29 )
Hence
t=x p
,
pr
TP
pr
y, z)
Tp
pr
Problem.
Under what conditions does the equation F(x,
?
=
define a devel
opable surface
We
u
assume that
x, y, z are
=
const, are the generators,
is
functions of two parameters w, u, such that the curves and v = const, are any other lines. The equation of
the tangent plane
This equation should involve u and be independent of
given by
(i)
u.
Its characteristic is
and
where we have put, for the sake
of brevity,
.t.^.^w+y, ax
2
_
dxdy
dxdz
dx
Since equation
(i)
is
independent of
u,
we have
(iii)
A* + B* + c dv dv
(ii)
= 0.
at)
Comparing equations
and
(iii)
with
(13),
we
see that
^X^=0, 3x
B\^=0, dy
CX^ oz
64
ENVELOPES
where X denotes a factor of proportionality. If we eliminate x, and X from these equations and (i), we obtain the desired condition
2
X
Y
y,
Z
z,
F F
d2
F F
2
d^F
dxdz
dz
dF
dx
z2
2
dxdy
d2
F
"
dF
dy
?
~fa
dx dy
dy
dy dz
az 2
=
0.
dx dz
d_F_
dy dz
aF
dy
**
dz
dx
EXAMPLES
1.
Find the envelope and edge of regression of the family of planes normal
Find the rectifying developable of a cylindrical
helix.
is
to
a given curve.
2.
3.
Prove that the rectifying developable of a curve
the polar developable of
its involutes,
4.
and conversely.
Find the edge of regression of the envelope of the planes x sin u au = 0. y cos u f z
Determine the envelope of a oneparameter family of planes parallel
line.
5.
to
a
given
6.
stant angle
Given a oneparameter family of planes which cut the xyplane under con the intersections of these planes with the latter plane envelop a curve C. Show that the edge of regression of the envelope of the planes is an
;
evolute of C.
7. When a plane curve lies on a developable surface its plane meets the tangent planes to the surface in the tangent lines to the curve. Determine the developable surface which passes through a parabola and the circle, described in a perpendicular plane, on the latus rectum for diameter, and show that it 4s a cone.
y2
Determine the developable surface which passes through the two parabolas z = 0; x2 = 4 ay, z = 6, and show that its edge of regression lies on the surface y*z = x 3 (6 z).
8.
= 4 ox,
the moving trihedral. Problems concern ing the envelope of a family of surfaces are sometimes more readily solved when the surfaces are referred to the moving trihedral of a curve, which is associated in some manner with
28. Applications
of
the family of surfaces, the parameter of points on being the parameter of the family.
the
curve
Let
(30)
F(&
77,
,
*)
=
APPLICATIONS OF THE MOVING TRIHEDRAL
define such a family of surfaces. Since f, 77, f are functions of the equations of the characteristics are (30) and
65
*,
^^^ + ^^ + ^^4.^=0 ~
~
ds
d% ds
dr)
ds
0f ds
ds
But the
characteristics being fixed in space,
we have
(I,
84)
Hence the equations
(32)
of the characteristics are
,_
,
/i
If,
for the sake of brevity,
we let $(,
??, f,
*)
=
denote the second
of these equations, the edge of regression
is
defined by (32) and
8S
&lt; &gt;
!(

For example, the family of osculating planes of a curve is defined with refer ence to the moving trihedral by f = 0. In this case the second of (32) is rj = 0, and
 4(33) is
=
0.
Hence the tangents are the
;
characteristics,
and the edge of regres
sion is the curve
for,
we have
= =f=
??
0.
In like
of .(32) is
17
manner
p=0
;
the family of normal planes is defined by = 0. Now the second consequently the polar lines are the characteristics. Equation (33)
;
reduces to f f p r = the edge of regression
hence the locus of the centers of the osculating spheres
18).
is
(cf.
The envelope
is
called the polar developable.
The
of surfaces
From
(
osculating spheres of a twisted curve constitute a family which is readily studied by the foregoing methods. 18) it follows that the equation of these spheres is
The second
of equations (32) for this case
is
which, since spherical curves are not considered, reduces to = 0. And equation (33) is ?? = 0, so that the coordinates of the edge of
regression are f
= = f = 0.
77
Hence
:
The osculating
lating spheres
;
circles
of a curve are the characteristics of
its
oscu
and
the curve itself is the edge of regression of the
envelope of the spheres.
66
ENVELOPES
29. Envelope of spheres.
Canal surfaces.
We
consider
now any
family of spheres of one parameter. Referred to the moving tri hedral of the curve of centers, the equation of the spheres is
By means of (32) we find that a characteristic which a sphere is cut by the plane
The
r=
is
the circle in
radius of this circle
is
acteristic
s
imaginary
equal to when r n
is
&gt;
rVl
1.
r
n
.
Hence
the char
1,
reduces to a point
when
+ const.,
and
is real for r
f
*
&lt;
By means
(34)
of (33)
we
by
find that the coordinates of the edge of
regression are given
f = n
,
, = [l(rr
)
]p,
r
two parts with
corre
Hence the edge
of regression consists of
sponding points symmetrically placed with respect to the oscu lating plane of the curve of centers (7, unless
When
is
this condition
is
satisfied the
edge
is
points lie in the osculating planes of C. the case with the osculating spheres of a curve.
We
a single curve, and its have seen that this
We
shall
show
that
their
when
the above condition
&lt;7
is
satisfied the spheres osculate
We
(35)
edge of regression r write the above equation in the form
p[l~(rr
e is
)]
^
= er^lr \
where
so that p may be positive. have seen ( 16) that the absolute and relative rates of change with s of the coordinates f, ?;, f of a point on Ct are in the relations or
1,
+1
We
M = ^_?? +
Ss
i,
ds
^ = ^Z + l + f,
&s
^
Ss
==
^_!?.
ds
T
p
ds
p
T
the values (34) are substituted in the righthand of these equations, we obtain, in consequence of (35),
When
members
ENVELOPE OF SPHEEES
Hence the
linear element Ss^ of
67
C
l
is
given by
cs 1
=
and
(36)
Uo,
Since these are the directioncosines of the tangent to C^ we see that this tangent is normal to the osculating plane to the curve of
centers C. Moreover, these directioncosines must satisfy the equations
/37\
(cf. I,
83)
8a
8s
_da
ds
b
8b
8s
_
db
ds
a p
c
8c
8s
do
ds
b
p
T
r
Hence we have
from which
(38)
it
follows that the radius of curvature p l of
Pl
C
l
is
=ee rVT^,
where
1 or 1 , so that /o 1 may be positive. Since, now, the is directioncosines of the principal normal have the values
e
r
+
it
follows that the principal normals to C and Cl are parallel. Furthermore, since these quantities must satisfy equations (37),
we have
g
3
g
3
,
^^
.
where
of
(I,
denotes the derivative of p l with respect to s r By means 51) we find that the radius of torsion r l of Cl is given by
p[
.From
(38)
we
find
p[=

so that the radius
R^ of the oscu
2 = r and consequently p? p[ TI lating sphere of C^ is given by R* the osculating spheres of Cl are of the same radius as the given
+
2
,
spheres.
68
ENVELOPES
The
directioncosines of the tangent, principal normal, and binor to Cl are found from (36) and (39) to be
mal
Hence the coordinates
of
Cl
of the center of the osculating (I, 94) sphere are reducible, in consequence of (34), to
+
l
iPi
 P( T I\ =
we have
*?
+ m iPi ~ friPi =
:
&gt;
? + W1p1
 XT^ = 0.
Therefore
the theorem
When
the edge of regression of a
family of spheres of one param
eter has only one branch, the spheres osculate the edge.
r
is
Since r does not appear in equation (35), it follows that when given as a function of s, the intrinsic equations of the curve of
are
where the function/(s) is arbitrary. Moreover, any curve will serve for the curve of centers of such an envelope of spheres. The deter mination of r requires the solution of equation (35) and consequently
involves two arbitrary constants. When all the spheres of a family have the same radius, the envelope is called a canal surface. From (34) it is seen that in
this case a characteristic is a great circle.
Moreover, equation (35)
a necessary and sufficient condition that the edge of regression of a canal surface consist of a single curve is that the curve of centers be of constant curvature and the radius
reduces to p
= r.
Hence
of the sphere equal to the radius of first curvature of the curve.
GENERAL EXAMPLES
1.
Let
MN be a generator of the right conoid
x
=
u cos u,
it
y
=
u
sin
i&gt;,
z
=
2 k cosec 2
D,
M being the point in which
2.
that the tangent plane at meets the surface in a hyperbola which passes through M, and that as moves to the hyperbola describes a plane. along the generator the tangent at
zaxis.
meets the
Show
N
M
N
A
2 a2 ft c2 motion always passes through the perpendicular from the center on the tangent plane at the point. Show that the path of the point is the curve in which the ellipsoid
point moves on an ellipsoid
h
H
 = 1,
so that the direction of
its
is
cut by the surface x l y m z n
=
const. ,
where
1:
m
:
n
 :
:

GENERAL EXAMPLES
3.
69
If
same angle about the tangent
each of the generators of a developable surface be revolved through the to an orthogonal trajectory of the generators at the
point of intersection, the locus of these lines is a developable surface whose edge of regression is an evolute of the given trajectory.
4.
Show
that the edge of regression of the family of planes
(1

w 2 )z
+
i(l
+
u*)y
+
2uz +f(u)
=
,
is
a minimal curve.
5.
The developable surface which passes through the
62 ,
x2
f
z2
6.
y
=
meets the plane x
=
in
circles x2 f y* = a 2 an equilateral hyperbola.
z
= 0;
surface az
Find the edge of regression of the developable surface which envelopes the = xy along the curve in which the latter is cut by the cylinder x2 = by.
ellipsoid
7. Find the envelope of the planes which pass through the center of an and cut it in sections of equal area.
8.
planes
The first and second curvatures ax + /3y + yz p, where /3,
&lt;r,
of the edge of regression of the family of 7, p are functions of a single parameter
u and a 2
+
2
/3
f
y
2
=
1,
are given by
1
A3 pp a
oc.
A2
where
aa O A= P P
a"
p
a"
f
P
off
,
D=
r\
p a
y
"
,
ft
y
9.
Y
p,
y"
p,,
"
Derive the equations of the edge of regression of the rectifying developable
of
28.
by the method
10. Derive the results of
11
.
29 without the aid of the moving trihedral.
circle
Find the envelope of the spheres whose diameters are the chords of a
latter.
through a point of the
12. Find the envelope and edge of regression of the spheres which pass through a fixed point and whose centers lie on a given curve.
13. Find the envelope and edge of regression of the spheres which have for diametral planes one family of circular sections of an ellipsoid. 14.
Find the envelope and edge of regression of the family of
&lt;j/2\
ellipsoids
1
H
j2

=
1,
where a
is
the parameter.
(3^2
15.
Find the envelope of the family of spheres whose diameters are parallel
ellipse.
chords of an
16.
Find the equations of the canal surface whose curve of centers is a circular helix and whose edge of regression has one branch. Determine the latter.
17.
Find the envelope of the family of cones
(ax
+
x
+
y
+
z

1)
(ay
+
z)

ax (x
+
y
+
z

1)
=
0,
where a
is
the parameter.
CHAPTER
III
LINEAR ELEMENT OF A SURFACE. DIFFERENTIAL PARAMETERS. CONFORMAL REPRESENTATION
30. Linear element.
Upon
y
a surface
,
defined by equations in
the parametric form
(1)
x =fi (i*,
v),
=/
a
(w, v),
2
=/, (M,
v),
we select any curve and write its equations $ (u, v) 0. From we have that the linear element of the curve is given by
(2)
3
d?
j foj = fa du  av,
t
where ax
\
^M
the differentials
&lt;^w,
^v
j ay
^ = dij au + dy
,
t
,
^w
vf
,
2
=
dz
dz , aw H,
9
du
dv
dv satisfying the condition
2$
^w
,
c?w
+ a^ dv =
,
0.
dv
ifweput
du dv
cu cv
du dv
or, in
abbreviated form,
equation
(4)
(2)
becomes
oV
2
= Edu? + 2 Fdudv + G dv
G
thus denned were
2
.
The
functions E, F,
the surface
first
used by Gauss.*
When
*
is real,
and likewise the curvilinear coordinates
Disquisitiones generates circa superficies curvas (English translation by Morehead Hiltebeitel), p. 18. Princeton, 1902. Unless otherwise stated, all references to Gauss are to this translation.
and
70
LINEAK ELEMENT
w, v, the functions
71
understand also that There is, however, an important excep tional case, namely when both E and G are zero (cf. 35). For any other curve equation (4) will have the same form, but the relation between du and dv will depend upon the curve.
are real.
Vj,
#
We shall
the latter are positive.
Consequently the value of c7s, given by (4), is the element of arc of any curve upon the surface. It is called the linear element of
the surface (cf.
20).
However, in order
to avoid circumlocution,
we
shall frequently call the expression for ds 2 the linear element,
is,
that
the righthand
member
of equation
(4),
which
is
also called
the first
ter,
fundamental quadratic form. The coefficients of the lat namely E, F, G, are called the fundamental quantities of the
for the sake of brevity,
first order.
If,
we put
du du
(5)
d(u, v) dv
it
dz
d(u, v)
d(u, v)
dv
follows from (3) and (5) that
(6)
EG  F = A + B + C
2
2
2
2
.
and likewise the parameters, is different from zero unless J, B, and C are zero. But if A, B, and C are zero, it follows from (5) that u and v are not independent, and consequently equations (1) define a curve and not a surface. However, it may happen that for certain values of u and v all the quantities J, B, C vanish.
the quantity
Hence when the
EGF
surface
2
is
real
The corresponding
face.
points are called singular points of the sur
These points may be isolated or constitute one or more curves upon the surface such curves are called singular lines. In the following discussion only ordinary points will be con
;
sidered.
From
the preceding remarks
it
follows that for real surfaces,
referred to real coordinate lines, the function
(?)
is real,
H defined
by
1
and
it is
positive
by hypothesis.
72
LINEAR ELEMENT OF A SURFACE
31. Isotropic developable.
and
H
is
zero, is afforded
The exceptional case, where the surface is imaginary by the tangent surface of a minimal curve. The equa
.
tions of such a surface are (cf
22)
w2
/I
(u)
du
+
1
w2
(M) u,
=
ju(f&gt;(u)
du
+
u&lt;j&gt;
(u) v,
where 0(w)
a function of u different from zero. It is readily found that J=v2 2 (w), F 2 = 0. This equation is likewise the sufficient and consequently EG that the surface be of the kind sought. For, when itjs satisfied, the equa condition 2 2 If X denote an tion of the linear element can be_written ds = (Vjdw + V(?dw) factor of \^Edu f V(?du, and a function MI be defined by the equation integrating
is
F= G
0,
.
\(\fEdu + VGdv)
dwi, the above equation
becomes ds2
=
A"**
duf.
Hence,
if
we we
const, and any other system for vi take for parametric curves u\ have FI = 0, GI = 0. In other form these equations are
=
const.
,
3v
In accordance with the last equation
we put
01?!
2
8i?i
where
By
undetermined. integration we have
A;
is
r
X, M, v
being functions of HI alone.
When
we
these values are substituted in the first
of the above equations of condition,
get
to be satisfied
by X, /A, and v. The equation of the tangent plane
(1
to the surface
(i)
is
reducible to

2
W]l
)
(X 
x)
+
i
(1
+
w
2
)(
Y
y)
+
2 Ml (Z

z)
=
0.
Hence the surface
Since its edge of regression is a minimal curve is developable. surface is called an isotropic developable. (Ex. 4, p. 69), the theorem is proved. The
32. Transformation of coordinates.
It is readily
found that the
functions E, F,
G
are unaltered in value
by any change of the
show that these functions rectangular axes. But now we shall there is a change of the curvilinear change their values when
coordinates.
TRANSFORMATION OF COORDINATES = u(u^ = v(u ,v
l
73
Let the transformation of coordinates be defined by the equations
(8)
u
Vj),
v
l );
then
we have
dx
du, i
_
dx du
dx dv
dv du. i
dx
dv. i
_ dx
du
du
dv,
du du, i
11
dv dv
dx dv
we
find the relations
fa
(9)
E
du dv
dv
dv. 1
du dv
dv, du,
11
d^
dv l
Hence the fundamental
forms when there
is
quantities of the first order assume a change of curvilinear coordinates.
differentiation,
new
From
(8)
we have, by
du
=
du
du.
11
du l
du
\
dv.
aVj,
dv
dv
du,
11
du.
+

dv
dv..
dv,
Solving these equations for du v dv^ we get
l/dv
where
(10)
,
du
,
\
1/
dv
,
.
du
d(u, v)
Hence we have
\
"du
du
8 dv l
dv
(du
so that
(12)
1
(9)
^
From
we
find the relation
LINEAR ELEMENT OF A SURFACE
By means
equations
of this equation
(9) into
and the relations
:
(11),
we can transform
the following
^^\^2F^
EG
E
(13)
l
+
G(
1
du
I*
0V,
fa to
1
&lt;
1 __
F (faito1
cu^ ct\
i
du
cu cu
EG
r^
,T
/"
1
cv
JG/Cr
Tfi S~1
CM
Jj&lt;2
1^1
33.
1
.T
WA
metric line v
in
Angles between curves. The element of area. Upon a para = const, we take for positive sense the direction
increases,
which the parameter u
u
= const,
we
the direction in which v increases.
the elements of arc of curves v
tively,
find,
and likewise upon a curve If efe,, and e?s M denote and u = const, respec const,
from
ds v
(4),
(14)
= ^Edu,
M,
ds u
= ^Gdv.
Hence,
if
a w #, y v and
,
/9M ,
tangents to these curves respectively,
y u denote the directioncosines of the we have
fa
du
1
cu
cy
"*&lt;/
E
du
dx
&lt;
1
dz
have seen that through an ordinary point of a surface there passes one curve of parameter u and one of parameter v. and 180, If, as in fig. 11, &) denotes the angle, between
We
formed by the positive directions of the tangents to these curves at the point, we have
(15)
cos
ft)
= aa +
,
+ 77 = 7==
W
and
(16)
sin
&)
= VJSQF*
H
ANGLES BETWEEN CUKVES
When
two families
of curves
75
upon
a surface are such that
through any point a curve of each family, and but one, passes, and when, moreover, the tangents at a point to the two curves through it are perpendicular, the curves are said to form an orthogonal system. From (15) we have the theorem:
A
necessary
and
sufficient condition that the
is that
a surface form an orthogonal system
F = 0.
parametric lines upon
Consider the small quadrilateral (fig. 11) whose vertices are the points with the curvilinear coordinates (u, v), (u f du, v),
du, v (u dv). To within terms of higher order the sides of the figure are equal. Consequently it is approxi opposite mately a parallelogram whose sides
(u, v
+ dv),
+
+
are of length
v E du and
is
o&gt;.
\
G dv and
The
area
the included angle
of this parallelogram is called the element of area of the surface. Its
expression
(17)
If
its
is
/(u+du.v)
FIG. 11
d^ = sin CD V EG dudv = H dudv.
C
is
any curve on a surface, the directioncosines tangent at a point have the form
dx __ _ _ /dx du
I
_
I _
a,
7 of
_
dx dv _
_
_
/dz
//
__
_.
__
dy ^
ds
I
,^
/dy ^4^
du
,
n
_
__1
ds
cu ds
dv ds
\cu ds
dy dv dv ds
g&lt;
_
J
If
~ dz _
du
cz
dv
~ds
ds~\^uds
dv
we put dv/du = X and the righthand member of
replace ds by the positive square root of (4), the above expressions can be written
dv
(18)
du
dv
7
=
du
dv
76
LINEAR ELEMENT OF A SURFACE
these results
it
From
is
not upon the
ratio X.
absolute values of
of
seen that the directioncosines depend du and dv, but upon their
obtained by differentiation from the
The value
(7,
X
is
equation of
(19)
namely
Let Cl be a second curve meeting C
the directioncosines of the tangent to
at a point
C
l
at
M
M, and
/3 V
let
be a v
yr
They
are given
by
"
l
du 8s
/3 t
dvW
8 indicates variation
and similar expressions
in the direction of
If 6
for
and 7^ where
Cr
C and
Cl
at
denotes the angle between the positive directions to M, we have, from (18) and (20),
cos
i
(21)
= #tf + ppj + 77j = Eduu + F(du 8v f dv 8u] +Gdv8v j^
x
and
sin 9
=
Vl  cos
is
=
H
dv
8s (8u ds
8v
du
8s ds
This ambiguity of sign
of
due to the
fact that 6 as
denned
is
one
two angles which together are equal to 360. upper sign, thus determining 6. This gives
/nft
.
We
take the
(22)
Q Tr sm6 = H
.
/8udv _
\09 d8
8v du\ __.
08 ds/
The
When
have 8v
/00 (23)
.
significance of the above choice will be pointed out shortly. const, through M, we in particular Cl is the curve v
=
.
=
and
~
8s
= V E 8u,
1
j
so that a
cos&lt;9
=
T,dv\ = { ^du + .F U ds
If dv
ds/
81X100==
From
(24)
these equations
we
obtain
tan*
=
Edu+Fdv
Hence there
is,
The angle
metric
co
between the positive half tangents to the para
uniquely denned.
in
curves has been
ANGLES BETWEEN CURVES
77
= const, general, only one sense in which the tangent to the curve v can be brought into coincidence with the tangent to the curve
u
=
const,
by a rotation of amount
co.
We
say that rotations in
this direction are positive, in the opposite sense negative. From is the angle described in the positive sense (23) it is seen that when the positive half tangent to the curve v const, is rotated
=
so in the general the angle described in the positive rota case 6, defined by (22), tion from the second curve to the first.
into coincidence with the half tangent to C.
is
And
From
(26)
equations (15), (16), and (23)
we
find
These equations follow also directly from (20) and
ering the curve
(21)
by consid
the
u = const,
as the second line.
As an immediate consequence
theorem
:
of equation (21)
we have
necessary and sufficient condition that the tangents to two curves upon a surface at a point of meeting be perpendicular is
(26)
A
E du Su + F(du Sv + dv 8u) + G dv $v = 0.
EXAMPLES
1.
Show
that
when
ds 2
the equation of a surface
is
of the
form z
2
)
=/(,
y), its linear
element can be written
=
(1
+ p2
)
dx 2
+ 2pqdxdy +
(l
+
q
dy 2
,
where p = dz/dx, and q = cz/dy. Under what conditions do the y = const, form an orthogonal system ?
2.
lines
x
=
const. ,
Show
that the parametric curves on the sphere
x
=
a
sin
u
cos
v,
y
=
a sin u sin
u,
z
=
a cos u
form an orthogonal system. Determine the two families of curves which meet the curves v = const, under the angles ir/4 and 3 7r/4. Find the linear element of the
surface
3.
when
these
new curves
are parametric.
y
Find the equation of a curve on the paraboloid of revolution x = wcosu, = w 2 /2, which meets the curves v = const, under constant angle a Determine a as a function of and passes through two points (M O i). (MI,
=
itsinu, z
,
i&gt;o),
4. Find the differential equation of the curves upon the tangent surface of a curve which cut the generators under constant angle a.
78
LINEAR ELEMENT OF A SURFACE
"
.
all 5. Show that the equations of a curve which lies upon a right cone and cuts cesinw, the generators under the same angle are of the form x = ce cosu, y the curve upon 2 = 6e", where a, 6, and c are constants. What is the projection of a plane perpendicular to the axis of the cone ? Find the radius of curvature of the curve.
6. Find the equations of the curves which bisect the angles between the para metric curves of the paraboloid in Ex. 3.
34. Families of curves.
An
equation of the form
(27)
&lt;(w,
v)=c,
infinity of curves, or a Through any point of the sur
where
c is
an arbitrary constant, defines an
family of curves, upon the surface. face there passes a curve of the family. For, given the curvilinear
coordinates of a point, we obtain a value of
when
c, say of the point. inquire whether this family passes through curves can be defined by another equation. Suppose it is possible, and let the equation be
;
CQ
these values are substituted in (27) then evidently the curve $ = c
We
(28)
^(U,V)
c
= K.
necessarily a function of the
if ifr is any same family of define the
Since
and K are constant along any curve and vary in passing
is
from one curve to another, each
other.
Hence
ir
is
a
function of fa
Moreover,
function of fa equations (27) and (28) curves. From equation (24) it is seen that the direction, at any point, of the the curve of the family through the point is determined by obtain the latter from the equation value of dv/du.
We
36
(29 )
,
_
d&lt;f&gt;
g*H.2*.i.&lt;*
which is derived from (27) by differentiation. Let $ (u, v) = cbe an integral of an ordinary of the first order and first degree, such as
(30)
differential equation
M(u,
v)
du
4
N(u,
v)
dv
= 0.
defined by the former equation are called integral curves of equation (30). From the integral equation we get equation (29) then to obtain equation (30) by differentiation. It must be possible
The curves
FAMILIES OF CURVES
from the integral equation and
(29).
79
does not appear in
But
c
(29), consequently the latter equation differs from (30) by a factor
at most.
Hence
M dv N du =
0.
Suppose, now, that we have
another integral of (30), as ^Hw,
v)
= e. Then
M ^ N ^ = dv cu
=
d(u,v)
o
.
o
,
0.
The
elimination of
it
M and N from these equations gives ^ ^
is
;
from which
follows that ty
&lt;.
a function of
&lt;/&gt;.
Moreover,
if
^r
can
But by any function of of the families of curves
&lt;,
we have seen
that
fy
= const,
&lt;/&gt;
and
= const,
^
is
a function
are the same.
Hence
admit
all integrals of
equation (30) of the form
&lt;f&gt;=c
or
^=e
may
obtain
define the
of
same family of curves.
c)
However, equation
this be solved for
(30)
an integral in which the constant of integration enters
implicitly, as F(u,
=
t&gt;,
0.
But
if
&lt;?,
we
one or more integrals of the form (27). Hence an equation of the form (30) defines one family of curves on a surface. Although
the determination of the curves
when thus
integration of the equation, the direction of is given directly by means of (24).
If at
defined requires the any curve at a point
each point of intersection of a curve
C with
l
the curves
of a family the tangents to the two curves are perpendicular to one another, Cl is called an orthogonal trajectory of the curves. Sup pose that the family of curves is defined by equation (30). The
7
r\
relation
between the
ratios
tions of the tangents to
is
du ou the two curves
If
and
&gt;
which determine the
r\
direc
at the point of intersection,
~~\/T
given by equation
(26).
we
replace
cu
by
A
0.
,
we
obtain
(31)
(EN FM) du + (FN
GM) dv =
But any integral curve of this equation is an orthogonal trajectory of the given curves. Hence a family of curves admits of a family of orthogonal trajectories. They are defined by equation (31),
when the differential equation of the curves is in the form But when the family is defined by a finite equation, such as
the equation of the orthogonal trajectories
is
(30).
(27),
(32)
80
LINEAR ELEMENT OF A STJKFACE
circles in the plane
As an example, we consider the family of on the xaxis whose equation is
(i)
with centers
x2
+
y2

2 ux

a2
,
where u is the parameter of the family and a is a constant. In order to find the orthogonal trajectories of these curves, we take the lines x = const. y = const.
,
for parametric curves, in
which case
E = G = 1,
and write the equation
(i)
F = 0,
thus
3?
in the
form
(27),
x
+
2
i (y 2 x

a2 )
=
2 u.
Now
equation (32)
is
2 xy dx
(x
y2
+ a2
)
dy
= 0,
of
which the integral
is
where v is the constant of integration. Hence the orthogonal whose centers are on the yaxis.
trajectories are circles
An
(33)
is
ordinary differential equation of the second degree, such as
H (u,
v)
du
2
+2
S(u,
v)
du dv
+ T(u,
v)
dv 2 =
0,
equivalent to two equations of the
first
degree, which are found
by solving this equation as a quadratic in dv. Hence equation (33) seek the con defines two families of curves upon the surface.
We
dition that the curves of one family be the orthogonal trajectories of the other, or, in other words, the condition that (33) be the equa
tion of an orthogonal system, as previously defined.
If
& x and
Jc
2
denote the two
values of
 obtained from du
(33),
we have
From
(26)
it
follows that the condition that the
two directions
is
at
a point corresponding to K I and K Z be perpendicular
E + FK +
If
tc
+ GK
= 0.
we have
the above values are substituted in this equation,
it is
the condition sought;
(34)
MINIMAL CURVES
35.
is
81
Minimal curves on a surface.
equating to zero
An
equation of the form (33)
obtained by
the
first
fundamental form of
a surface.
This gives
Edv?+ ZFdudv + Gdv2 =
and
it
0,
zero which
to JSG
defines the double family of imaginary curves of length In this case equation (34) reduces lie on the surface.
F =0;
2
hence the minimal lines on a surface form an
orthogonal system only
able
(
when
the surface
is
an isotropic develop
31).
important example of these lines is furnished by the system on the sphere. If we take a sphere of unit radius and center at
the origin, the forms
its
An
equation, x
2
+ y*+ z = l,
2
can be written in either of
1
z
x
f
iy
where u and
v denote the respective ratios,
and evidently are conju
?/,
gate imaginaries. If these four equations are solved for z,
z,
we find
u
s**
,
h v
Ai
i
.,
i(v \
u)
f_
A
uv
,
1
9
~uv+l
From
these expressions
uv
+\
uv+1
we
find that the linear element, in terms
of the parameters
u and
v^ is
given by
,o
(36)
4:dudv
(1
+
v
Hence the curves u
zero.
= const,
and
const, are the lines of length
Eliminating u from the
(35),
first
two and the
v*)y
last
two of equations
we get
(37)
Hence
all
the
= 0, = 0. i(i? + l)z + 2vy+i(lv points of a curve v = const, lie on the
x
4 (1
2 iv
z
1
)
line
v
z
)Y
2iv
= Q,
82
LINEAR ELEMENT OF A SURFACE
F,
where X,
Z denote
current coordinates.
In consequence of
(35),
these equations can be written
Xx,
,
Yy.
Zz,
point.
where # y^ z are the coordinates of a particular manner the curves u = const, are the minimal lines
In like
Xx.
Yy. = Zz,
,
EXAMPLES
1
.
Show
that the most general orthogonal system of circles in the plane
34.
is
that
of the
2.
example in
Show
(w
2
that on the right conoid x
dw 2
3.
f
a 2 ) dv 2
=
= ucosv, y form an orthogonal system.
ds*
=
usinv, z
=
au, the curves
When the coefficients of the linear ds 2 = Erfu* + 2 Fidudv + Gidv 2
,
elements of two surfaces,
= E 2 du? + 2 F2 dudv + G 2 di; 2
,
are not proportional, and points with the same curvilinear coordinates on each of the surfaces are said to correspond, there is a unique orthogonal system on one
surface corresponding to an orthogonal system on the other;
its
equation
0.
is
(Fi^a
4.
2
FzEi)du*
+ (E a Gi
 EiG z
)
dudv
2 +(GiFz  G2 Fi)du =
If 61
and
are solutions of the equation
a^
At/
dag/3
2
/J,
da
_ u,
8?
where X
is
any function
of
a and
the equations
+*5
define a surface referred to its
3 2*
.
minimal
lines.
36.
Variation of a function.
?/,
system of coordinates
v,
Let S be a surface referred to any and let (w, v) be a function of u and v.
&lt;j&gt;
When
the values of the coordinates of a point
&lt;,
Tlif
o f the surface are
substituted in
(38)
we
obtain a
number
c
;
and consequently the curve
VABIATION OF A FUNCTION
83
along this curve the passes through M. In a displacement from remains the same, but in any other direction it changes value of
(f&gt;
M
and the
rate of
change
is
given by
d
d$ k
dv
dc)
du
where k
dv/du determines the
direction.
As
thus written
it is
is
understood that the denominator of the righthand member
positive.
For the present we consider the absolute value of ~t and write
ds
du
(39)
dv
ds
e is
where
negative. direction along the curve (38). In order to find the maximum value we equate to zero the derivative of A with respect to Jc. This gives
1 according as the sign of the numerator is positive or The minimum value of A is zero and corresponds to the
From
(32)
it
follows that this value of k determines the direction
at right angles to the tangent to $ this value of k in (39) we get the
= c at the
point.
By
substituting
maximum
&lt;f&gt;
value of A.
(w, v) at
Hence:
The differential quotient ^ of a function ds
surface varies in value with the direction
zero in the direction tangent to the curve
absolute value in the direction
a point on a
&lt;f&gt;=
from the point. It equals c, and attains its greatest
value being
normal
to this curve, this
4
&lt;
&gt;
;
S
m^\ZFzz+G dv du dv
A means of
ential quotient
representing graphically the magnitude of the differ A for any direction is given by the following theorem
to
:
If in the tangent plane
tangents at
a surface at a point
to all
M
the positive half
M, corresponding
values of
k, positive
and
negative,
84
be
LINEAR ELEMENT OF A SURFACE
drawn, and on them the corresponding lengths
const.
A be
laid off from
M,
the locus of the extremities of these lengths is a circle tangent to the
curve
&lt;=
The proof
of this
theorem
is
simplified
if
we
effect a transfor
mation of curvilinear coordinates. Thus we take for the new coor const, and their orthogonal trajectories. dinate lines the curves
(f&gt;
We
vl
let the
former be denoted by u v
const,
and the
latter
by
= const.,
and indicate by subscript
1 functions in terms of these
parameters.
Now F = 0,
l
t J.
so that
7
where
\
direction,
denotes the value of dvjdu^ which determines a given and the maximum length is (J&\)~*. From (23) we have
cos
=
.
sin
=
where 6 Q
is
the angle which the given direction makes with the
const. Hence if we regard the tangents tangent to the curve v l const, as axes of coordinates const, and u l at to the curves v l
=
M
=
in the tangent plane, the coordinates of the end of a segment of length A are
distance from this point to the midpoint of the = segment, measured along the tangent to v t const.,
The
maximum
is
found to be

readily
=&lt;&gt;
which proves the theorem.
first order.
37. Differential
parameters of the
If
we put
(41)
A^ =
equation (40) can be written
(3)
where now the
normal to the curve
&lt;&gt;
**
differential quotient corresponds to the direction The lefthand member of this const.
=
DIFFERENTIAL PARAMETERS
equation
is
85
evidently independent of the nature of the parameters
u and
is
which the surface is referred. Consequently the same true of the righthand member. Hence A^ is unchanged in
v to
value
when
there
is
any change
,
of parameters whatever.
The
set of
full significance of this result is as follows.
Given a new
,
parameters defined by M=/I (M I i^), v=/2 (w 1 v^\ let ^(u^ vj denote the result of substituting these expressions for u and v in (w, v), and write the linear element thus
&lt;
:
ds 2
=E
l
du* +
2F
l
du l dv l + G
l
dv*.
The
invariance of
A^
under
,
this transformation
is
expressed by
the identical equation
EGF
We
leave
it
to the reader to verify this directly with the aid of
equations
(9).
The
invariant
A^
is
ter of the first order ; this
name and the
called the differential parame notation are due to Lame.*
Consider for the
(42)
moment
the partial differential equation
A^ =
&lt;/&gt;
and a solution
= const. From the latter we get, by differentiation,
d&lt;l&gt;
,  du
f
,
, 3$ dv
A = 0.
du
OJ O
J
dv
in (42)
If
we
replace
and
by dv and
du,
which are evi
dently proportional to them,
we
obtain
Edu*+
Hence the
is
2
Fdudv
+ Gdv*= 0.
integral curves of equation (42) are lines of length zero, = const, is a line of length zero, the function and conversely if
(/&gt;
&lt;/&gt;
a solution of equation (42). Another particular case is that in which
A^ is a function of
&lt;,
say
(43)
*
A,*
les
=
*&lt;*).
Lemons sur
coordonnees curvilignes
et leurs diverses applications, p. 5. Paris, 1859.
86
LINEAR ELEMENT OF A SURFACE
(41)
it is
From
seen that
when we put
equation (43) becomes
(44)
A
denned, 6
is
1
l9=l.
As
6
const, is
a function of $; hence the family of curves = const. Suppose we have the same as the family
&lt;/&gt;
const, for the curves such a family, and we take the curves 9 u = const, and their orthogonal trajectories for v = const., thus it follows from effecting a change of parameters. Since Aj%=l,
(41) that
(45)
^ = 1,
and consequently the linear element
ds
2
is
=du? + Gdv
2
.
Since
now
the linear element of a curve v
const,
is
du, the length
of the curve
between
its
u
=u
and u
= u^
const, between these two curves. the segment For this reason the latter curves are said to be parallel. Con = const, of an orthogonal sys versely, in order that the curves u
u^ of every curve v
is
UQ
points of intersection with two curves Moreover, this length is the same for
.
tem be
curves v
that the linear parallel, it is necessary
element of the
must be a func const, be independent of v. Hence of coordinates, can be tion of u alone, which, by a transformation made equal to unity. Hence we have the theorem
:
=
E
A
=
(/&gt;
the curves of a family necessary and sufficient condition that be a function of const, be parallel is that
\(f&gt;
&lt;f&gt;.
be the equations of two curves upon a surface, through a point M, and let 6 denote the angle between the tangents at M. If we put
Let
(f)
= const,
and
^ = const,
(46)
\(*,*)=
E
_F
dv dv
\dv du
+
G
du dv
/
du du
EG
I"
the expression (21) for cos 6 can be written
cos ,
(47)
DIFFERENTIAL PARAMETERS
an invariant for transformations of coordinates, follows from this equation that A x (0, ^r) also is an invariant. It
Since cos 6
is
87
it is
called the mixed differential parameter of the first order. diate consequence of (47) is that
An imme
A,(*.
is
i/r
*=
)
the condition of orthogonality of the curves
= const,
(f&gt;
and
= const. Now equation
(22) can be written
r
\du dv
dv c
which by means of the function
&lt;e)
(w, v),
defined thus by Darboux,*
can be written in the abbreviated form
(49)
sin 6
=
^r)
the functions in this identity except to be invariants, we have a proof that it also
all
is
Since
are
(&lt;^,
known
It
is
an invariant.
a
mixed
(49)
it
differential
parameter of the
2
((/&gt;,
first
order.
From
(47)
and
(50)
follows that
2
A,
(&lt;,
f+
)
t)
 A^
A^
;
consequently the three invariants denned thus far are not inde pendent of one another.
From
(41)
and
(46)
ri
it
follows that
^U =
and from these we
2
W
find
A
^ ^=l^
_
Tf&lt;
AlV
=
^
jfi
(51)
(u, v)
= A,H
VA
2
2
i
(.
)
=
^
first order.
Consequently
@
Hence i,
2
(w,
1
t;)
(w, v)
^, and G are differential invariants of the
* Lemons, Vol.
Ill, p. 197.
88
LINEAR ELEMENT OF A SURFACE
Another result
of these equations is the following. If the param changed in accordance with the equations
Ui
eters of the surface are
= Ui(u,
v),
v^v^u,
is
v),
and the resulting linear element
ds*
written,
=E
l
du* + 2 Fl du^dv^ + 6^ dv*,
the value of
E
l
is
given by
and 7^ and 6^ are found
in like
manner.
In consequence of (51)
these equations are equivalent to (13), which were found by direct calculation.
Thus far we 38. Differential parameters of the second order. have considered differential invariants of the first order only. We introduce now one of the second order, discovered by Beltrami.*
To
this
end we study the integral
n=
for an ordinary portion of the surface
(cf.
bounded by a closed curve C
33).
For convenience we put
(53)
M=
G *
z_ F du

d
dv
,
N=
we have
dv
E z_ F 3
du
,
so that, in consequence of (46),
This
may
be written
If we apply Green reduces to
s
theorem to the
first integral, this
equation
(54)
n=
(
C(j&gt;(MdvNdu)
ff$(^+ j^}
dudv
&gt;
*Ricerche di analisi applicata
(1864), p. 365.
alia geometria, Giornale di matematiche, Vol. II
DIFFERENTIAL PARAMETERS
where the
first
89
integral
is
curvilinear and
is
taken about
C
in the
Evidently du and dv refer to a displacement C. If we indicate by 8 variations in directions normal to C along and directed toward the interior of the contour, then from (23)
customary manner.
and
(25) it follows that
Edu + Fdv _H8v
F du + G dv __
ds
8s
~dT
~ST
Hence
M dv
du Ss
dv 8*/
8s
All of the terms in this equation, with the exception of

H\du
(
h r
dv J
)
&gt;
are independent of the choice of parameters. Hence the latter is an invariant. It is called the differential parameter of the second
order and
is
denoted by
A
2 i/r.
In consequence of (53) we have
.....
(56)
In the foregoing discussion
quantities appear.
But
all
has been assumed that only real these results can be obtained directly
it
from algebraic considerations of quadratic differential forms * without any hypothesis regarding the character of the variables hence the differential parameters can be used for any kind of
;
curvilinear coordinates.
In addition to
A
2
there are other differential invariants of the
c/&gt;
second order, such as
And
are
find a
AA Q,
i/r),
A, (A^,
A^),
(A
mixed invariants
of the second order.
In like manner
;
we can
group of invariants of the third order
AAM&gt;,
AA(4&gt;.M&gt;)&gt;
for instance,
A.A,*,
A A*.
I,
* Cf. Bianchi, Lezioni di geometria differ enziale, Vol.
chap.
ii.
Pisa, 1902.
90
LINEAR ELEMENT OF A SURFACE
others,
These invariants and
their derivatives.
extension of this method, involve functions
which can be obtained by an evident E, F, G, and A/T,
.
c/&gt;,
,
Conversely,
T ftw /== /(A
we
shall
show
,
* that every invariant of the
form
v ^
.
.
r dE
tr,
dG
,
di&gt;
*W
dw
dw
0, ^,
I
,
i/r,
^L,
^
du
),
where
of the
.
&lt;,
^,
are independent functions,
is
symbols
A
and
.
Already we have seen
expressible by means that E, F, and G
can be expressed in terms of
(48)
it
A
x
w,
A
x v,
and
A
1
(w, v).
Moreover, from
follows that
when X
any function whatever. Hence expressed in terms of the symbols A and
is
all
,
the terms in
/ can be
applied to
Since u and v do not appear explicitly in
of parameters, replacing
/,
we can
,
effect a
u and
v
by
and
&lt;/&gt;
ty respectively,
change and con
to these
sequently
we
express
/
in terms of
(/&gt;,
^,
and the
differential
&lt;*)
invariants obtained by applying the operators A and functions. In case $ is the only function appearing in
/,
c/&gt;,
we can
such as
take for
i/r,
2
&lt;,
in the
change of parameters, any invariant of
it is
A^
or
A
so long as
not a function of
(/&gt;,
E,
F&gt;
or G.
EXAMPLES
4
1
.
When
is
the linear element of a surface
is
in the
dv^),
form
ds 2
=
\(du^
+
where X
both u and v are solutions of the equation A 2 = the differential parameter being formed with respect to the righthand member.
a function of u and
D,
0,
2.
Show
that on the surface
x
the curves
3.
it
=
u cos u,
y
= u sin v,
form
z
=
av
f
(u),
=
const, are parallel.
is
When
is
the linear element
in the
ds2
=
u,
cos^adu 2
+
sin 2
a: eh? 2 ,
where a
a function of u and
both u and v are solutions of the equation
* Cf. Beltrami,
I.e.,
p. 357.
SYMMETRIC COORDINATES
4. If the
91
curves
the projection
= const., \p = const, form an orthogonal system on a surface, on the xaxis of any displacement on the surface is given by
dx
dx =
d\b
=
+
dx
*
dd&gt;
2=
A0
,
where ds and
respectively.
5.
da
are the elements of length of the curves
=
const.,
^
=
const.
If
/and
are any functions of u and
u,
then
.
,
0)
=
a/ ^ a0 ^ du AIU du
2
i if a0 /3A A + (^  +   A! (w,
.
,
u)
+
\cu cv
2
cv du/
a/a0. ^^ cv cv
v)
Aii&gt;,
A2 / =
39.
^A w + ^A u + ^A lW + 2^ Ai(u, SU
CM
CU
2
0ttCtJ
+ ^AiU. SV 2
Symmetric
co drdinates.
We
have seen that through every
point of a surface there pass two minimal curves which lie entirely on the surface, and that these curves are defined by the differential
equation
Edv?+2 Fdudv + G dv =
2
0.
If the finite equations of these
curves be written
fi (w, v)
a (w,
it
= const., v)
A, ()=&lt;),
=
const.,
follows from (42) that
(5T)
A
1
(/3)
= 0.
Since for any parameters
/^\
w=
when the curves a ~ const., ft = const., are taken as parametric, the corresponding coefficients and G are zero, and consequently the linear element of the surface has the form
E
(59)
ds
2
= \ dad/3,
where, in general, X is a function of a and {3. Conversely, as fol lows from (58), when the linear element has the form (59) equa
tions (57) are satisfied
Hence the only transformations
form of the linear
parametric, that
(60)
is
and the parametric curves are minimal. of coordinates which preserve this element are those which leave the minimal lines
a
or
=

92
LINEAR ELEMENT OF A SURFACE
where
F and F
l
are arbitrary functions.
Whenever
:
the linear ele
ment has the form (59), we say that the parameters are symmetric. The above results are given by the theorem
are symmetric coordinates of a surface, any two coordi arbitrary functions of a and ft respectively are symmetric nates, and they are the only ones.
When a and
ft
The
as the general linear element of a surface can be written
product of two factors, namely
(61)
d**:
If
denote integrating factors of the respective terms of the of symmetric coordinates a righthand member of this equation, pair
t
and
t
1
is
given by the quadratures
(62)
When
these values are substituted in (61), and the result
(59), it is
is
com
pared with
seen that X
=
tt
l
The
first
of equations (62) can be replaced by
,
= da
du
^FiH
&gt;
da&gt;
t
Eliminating
t
from these equations, we have
E^F^ du dv
63
&lt; &gt;
IT
equation be
=l
:
.
ccc,
Tu
~
If
Z
this
multiplied by
the
result
can be
reduced to
r*o cu dv
.
dec
ISOTHERMIC PARAMETERS
From
these equations
it
93
follows that
or,
by
(56),
(65)
It
is
A
readily found that
/3
2
tf
=
0.
also satisfies this condition.
40. Isothermic
real,
and isometric parameters.
also,
When
the surface
is
and the coordinates
imaginary. Hence for t r In this case a and
the factors in (61) are conjugate the conjugate imaginary of t can be taken
are conjugate imaginary also. In that this choice has been made, and write
fi
what follows we assume
(66)
a
=
&lt;+ty,
=
&lt;/&gt;
iyfr.
If these values be substituted in (59),
we
get
(67)
ds* =
\(d&lt;t&gt;
2
At once we
see that the curves
+d^). = const,
c/&gt;
and
^r
= const,
form
an orthogonal system.
lines are V\d*fr
Moreover, the elements of arc of these
increments
(&lt;f)
d&lt;f&gt;
and respectively. Consequently when the and d^ are taken equal, the four points i/r),
^\d(f&gt;
(&lt;,
f
c?&lt;,
i/r),
($,
i/r
f
efo/r),
f
(&lt;
tity,
= const, and small square. Hence the curves the surface into a network of small squares.
(f&gt;
^ f
eityr)
are the vertices of a
^r
= const,
this
divide
On
and
account
these curves are called isometric curves, and
&lt;f&gt;
ty isometric
parameters.
These lines are of importance in the theory of heat, and are termed isothermal or isothermic, which names are used
in this connection as
synonymous with
isometric.
the linear element can be put in the symmetric form, equations similar to (66) give at once a set of isometric parameters. And conversely, the knowledge of a set of isometric parameters leads
at once to a set of
Whenever
symmetric parameters. But we have seen that when
(60).
one system of symmetric parameters
given by equations of the form
known, all the others are Hence we have the theorem
is
:
&lt;,
Given any pair of real isometric parameters
every other pair
&lt;
v/r
for a surface
;
x,
ty 1
is
given by equations of the
form
to
where
F and F
Q
are
any functions conjugate imaginary
one another.
94
LINEAR ELEMENT OF A SURFACE
Consider, for instance, the case
(68)
*
1
+^
1
= ^(0+i.
From
(69)
the CauchyRiemann differential equations
?*i
d&lt;l&gt;
= *i,
c^
?& = _?*i,
3^
8&lt;f&gt;
&lt;
it
follows that
&lt;/&gt;
(f) l
and
the curves
&lt;f&gt;
1
const.,
i
T/T
= const., ^ = const, are = const. Similar results
1
^
l
are functions of both
different
and
T/T.
Hence
replaced
from the system
by
in the
is
argument
of the
when righthand member
hold
+i
is
of (68).
Hence
There
face; when one system
If the value (66) for
a double infinity of isometric systems of lines upon a sur is known all the others can be found directly.
a be substituted in the
is
first of
equations (57),
the resulting equation
reducible to
Since
&lt;f&gt;
and
^r are real, this equation is equivalent to
(70)
A^A.VT,
(58)
it is
A
1
(^,f)
= o.
From
E
G,
F= 0,
when
seen that these equations are the condition that and i/r are the parameters. Hence equations
&lt;f&gt;
(70) are the necessary and sufficient conditions that isometric parameters.
$ and
all
i/r
be
Again, when a in (65) tions are real, we have
(71)
2
is
replaced by
&lt;/&gt;+
i^r,
and
f
the func
A *=0,
when we have
a function
(f&gt;
Conversely,
satisfying the first of these
equations, the expression
cu
cv
,
on
ov
dv
is
,
an exact
differential.
Call
it
d^r
;
then
jr^
du
_
E
G
du
~~du
F
c)v
H
dv^_c^r
H
ISOTHERMIC ORTHOGONAL SYSTEMS
If these equations
95
be solved for
du
dv
we get
/r
_
ox
dv
du
(&lt;o)
H
=
dd&gt;
dv
"&gt;
du
du
H
(
)
=
d&lt;f&gt;
dv
When we
du\dvj and ^ satisfy (70), A 2 /r=0. Moreover, these two functions in consequence of (72) and (73), and therefore they are isometric
dv\duj
(f&gt;
express the condition
=
&gt;
(
)
we
find that
parameters.
Hence
:
A
the
necessary and sufficient condition that
be the isometric
&lt;f&gt;
param
c/&gt;
eter of one
family of an isometric system on a surface is that A 2 = isometric parameter of the other family is given by a quadrature.
;
Incidentally
we remark
that
if
u and
v are a pair of isometric
(69).
parameters, equations (72)
and
(73) reduce to
41. Isothermic orthogonal systems.
If the linear
element of a
surface
is
given in the
form (67) and the parameters are changed
in accordance with the equations
the linear element becomes
where the accents indicate
differentiation.
However,
this trans
;
formation of parameters has not changed the coordinate lines the coefficients are now no longer equal, but in the relation
&lt;&gt;
if
U and V denote
functions of u and v respectively. this relation is satisfied the linear element
where
may
Conversely, when be written
and by the transformation of coordinates,
(75)
=
4&gt;
C^/lfdu,
^= C
96
it is
LINEAR ELEMENT OF A SURFACE
brought to the form
(67),
whatever be
:
U and
V\ and the coor
dinate lines are unaltered.
Hence
necessary and sufficient condition that an orthogonal system of parametric lines on a surface form an isothermic system is that the
coefficients
A
of the corresponding linear element satisfy a relation of
the
form
(74).
We
and
Either
curves
seek
o&gt;
now
function
(w, v)
must
the necessary and sufficient condition which a = const. satisfy in order that the curves
o&gt;
their
o&gt;,
orthogonal
trajectories
it,
or a function of
is
form an isothermic system. the isothermic parameter of the
then
&lt;/&gt;;
o&gt;
= const. We
A
o&gt;
denote this parameter by
.
&lt;/&gt;=/()
Since
&lt;/&gt;
must be a solution
2
of equations (71), we have, on substitution,
(G&gt;)
(76)
./
+
\a&gt;
./"(a&gt;)
= 0,
to
&&gt;.
where the primes indicate differentiation with respect equation is written in the form
If this
we
see that the ratio of the
two
differential parameters
o&gt;,
is
a func
tion of
/(a&gt;),
co. Conversely, if this ratio is a function of obtained by two quadratures from
the function
(77)
/
necessary and
const,
is
(*&gt;)
= */&gt;,
condition
that
will satisfy equations (71).
Hence:
a family of curves
isothermic sys
&&gt;.
A
a)
sufficient
=
and
their orthogonal trajectories
form an
tem
that the ratio of
A
&&gt;
&&gt;
2
and
AjO) be
a function of
;
a function w then the orthogonal tra = const, can be found by quadrature ; for, the curves jectories of the differential equation of these trajectories is
Suppose we have such
(78 ^
If

"

\~
dv
du/
\~ dv
Su
equation (76) be written in the form
I
*f&gt;
dv
TT
du
H
*
v
r\
i
,
.
wu
TT
t/i/
O
&lt;7V
I
*^
f
(&)) \ /
I
I
4/
r
(ft)) \ /
n = u,
ISOTHEEMIC ORTHOGONAL SYSTEMS
it is
97
seen that an integrating factor of equation (78) is f (a))/H, where f (co) is given by (77). Hence /() and the function obtained by the quadrature
__#&&gt;
&lt;f,
^cto
s^fo
rJo&lt;*
are a pair of isometric parameters. it follows that
From
these equations and (77)
and consequently, by means given the form
(80)
ds*
of (52), the linear element can be
d + /sS ctyA
8
"
=
A^
(da* x
The
2 linear element of the plane referred to rectangular axes is ds
=
dx 2
:
4
dy 2
.
Consequently x and y are isothermic parameters, and
we have
the theorem
The plane curves whose equations are obtained by equating to constants the real and imaginary parts of any function of x + iy or x  iy form an isothermal orthog onal system ; and every such system can be obtained in this way.
For example, consider
where
c is
4
ty
c = 2
x
iy
any constant. From
this
it
follows that
x2
4
yz
x2
4
y
2
= const., $ = const, form an isothermal orthogonal system, Hence the circles and and ^ are isothermic parameters. The above system of circles is a particular case of the system considered in 34. We inquire whether the latter also form an isothermal system. If we put
u
1
= x 4(w
2
i
(2/
2

2
),
we
find that
AIO&gt;
=
4
x2
4 a 2 ),
A^u
=
2d)
x2
Hence the
ratio of
AIW and A2W
circles is isothermal.
first
From
tan 1
a function of w, and consequently the system of follows that the isothermic parameter of the (77)
is
it
family
is
=
2a
,
2a
1
\b
and the parameter of the orthogonal family
is
tanh 1
w
&gt;
w
2a
2a
=y
x2
4
4
a2
y
98
LINEAR ELEMENT OF A SURFACE
EXAMPLES
that the meridians and parallels on a sphere form an isothermal orthog onal system, and determine the isothermic parameters.
1.
2.
Show
Show
Show
that a system of confocal ellipses and hyperbolas form an isothermal
orthogonal system in the plane.
3.
that the surface
2
x
a
is
_ ~
I
(a

u)
2
)
\ (a  &
2
(a^v) 2  c2 (a
y
b
_ ~
I
(b*

u) (b
2 2
)
\ (62 
a2 )
(6

v) c2 )
z
c
_ ~~
!
(
C2
2
_
U)
(
C2
_
2
)
\f(c
a2 )(c2 6
an
4.
ellipsoid,
and that the parametric curves form an isothermal orthogonal system.
the surface
Find the curves which bisect the angles between the parametric curves on %_u+v y_u_v _ uv
a~
=
2
b~"~2~
2
and show that they form an isothermal orthogonal system.
5. Determine u cos v, y u sin v, z = (v) (v) so that on the right conoid x the parametric curves form an isothermal orthogonal system, and show that the curves which bisect the angles between the parametric curves form a system of
&lt;f&gt;
=
the
same kind.
6.
Express the results of Ex.
4,
page 82, in terms of the parameters
and ^
defined
by
(66).
42. Conformal representation.
ence of any kind
is
a onetoone correspond established between the points of two sur
When
faces, either surface
may
be said to be represented on the other.
Thus,
if
we
roll
out a cylindrical surface upon a plane and say
that the points of the surface correspond to the respective points of the plane into which they are developed, we have a representa
upon the plane. Furthermore, as there is no or folding of the surface in this development of it upon stretching the plane, lengths of lines and the magnitude of angles are unal
tion of the surface
a representation of every surface upon a plane, and, in general, two surfaces of this kind do not admit of such a representation upon one another.
tered.
It is evidently impossible to
make such
However,
it is
possible, as
we
shall see, to represent one surface
upon another in such a way that the angles between correspond ing lines on the surfaces are equal. In this case we say that one
surface has conformal representation on the other. In order to obtain the condition to be satisfied for a conformal
representation of two surfaces S and S we imagine that they are referred to a corresponding system of real lines in terms of the
r
,
CONFOKMAL KEPKESENTATION
same parameters
respective forms
w,
v,
99
and that corresponding points have the same
curvilinear coordinates.
We
2
write their linear elements in the
ds
2
= Edu*+ 2 Fdudv + G dv
to
,
ds
2
=
du*+
2
F dudv + G
dv*.
co and must be sponding points
Since the angles
between the coordinate
it is
lines at corre
equal,
necessary that
F
(81)
F
y/EG
and
Q
r
denote the angles which a curve on S and the corre sponding curve on S respectively make with the curves v = const. at points of the former curves, we have, from (23) and (25),
If
.
sin
n
=
,

H
dv
,
.
.
sin
ds
a sm 6 =
.
(to
 Q = H du =
)
H=, dv
s
sin
,
, 1
(to
H ai\  6[) = = du
V
&lt;&?
By
r
hypothesis
a&gt;
=a)
~
and
6[
Q,
according as the angles have
the same or opposite sense.
Hence we have
H
H
ds
~
H
du
according to the sense of the angles.
From
these equations
we
find
which, in combination with (81),
may
be written
where
2
t
v in general.
denotes the factor of proportionality, a function of u and From (83) it follows at once that
ds
(84)
*=t *
And
and
so
when
(82) follow.
the proportion (83) is satisfied, the equations (81) Hence we have the theorem
:
necessary and sufficient condition that the representation of two surfaces referred to a corresponding system of lines be conformal is
A
100
that the first
LINEAR ELEMENT OF A SUKPACE
fundamental
coefficients
of the two surfaces be propor tional, the factor of proportionality being a function of the param eters ; the representation is direct or inverse according as the relative
positions of the positive half tangents to the parametric curves on the two surfaces are the same or symmetric.
Later
we
shall find
it
From
(84)
means of obtaining conformal representations. follows that small arcs measured from correspond
f
ing points on S and S along corresponding curves are in the same ratio, the factor of proportionality being in general a function of
the position of the point. Conversely, when the ratio is the same for all curves at a point, there is a relation such as (84), with t a function of u and v at most. And since it holds for all directions,
we must have the proportion (83). On this account we may say that two surfaces are represented conformally upon one another when in the neighborhood of each pair of homologous points corre
sponding small lengths are proportional.
par equal to unity, corresponding small lengths are equal as well as angles. In this case the representation is said to be isometric, and the two surfaces are said to be applicable. The
ticular the factor
t is
43.
Isometric representation.
Applicable surfaces.
When in
significance of the latter term is that the portion of one surface in the neighborhood of every point can be so bent as to be made to
coincide with the corresponding portion of the other surface with out stretching or duplication. It is evident that such an applica tion of one surface upon another necessitates a continuous array of
surfaces applicable to both S and $r This process of transformation is called deformation, and Sl is called a deform of S and vice versa.
An
example of
this is afforded
by the rolling of a cylinder on
a plane.
Although a conformal representation can be established between any two surfaces, it is not true, as we shall see later, that any two surfaces admit of an isometric representation upon one another. From time to time we shall meet with examples of applicable sur faces, and in a later chapter we shall discuss at length problems which arise concerning the applicability of surfaces. However,
we
consider here an example afforded by the tangent surface of a
twisted curve.
APPLICABLE SURFACES
101
on the
We
of the
recall that if #, y, z are the coordinates of a point
curve, expressed in terms of the arc, the equations of the surface are
form
f
=x+
^
v
=y+y
t,
=z + z%
and the linear element of the surface
is
d
&lt;r*
= /I + \ ds +
z
2 dsdt
+
dt\
where p denotes the radius of curvature
of the curve.
Since this expression does not involve the radius of torsion, it follows that the tangent surfaces to all curves which have the same intrinsic equation p =f(s) are applicable in such a way that
points on the curves determined by the same value of s correspond. As there is a plane curve with this equation, the surface is appli cable to the plane in such a way that points of the surface corre
spond to points of the plane on the convex side of the plane curve.
to a curve are the characteristics of the osculating planes as the point of osculation moves along the curve, and con
The tangents
it
sequently they are the axes of rotation of the osculating plane as moves enveloping the surface. Instead of rolling the plane over
the tangent surface, we may roll the surface over the plane and bring all of its points into coincidence with the plane. It is in this sense
that the surface
it is
is
called a developable surface
developable upon a plane, and for this reason (cf. 27). Later it will be shown
is
that every surface applicable to the plane a curve ( 64).
the tangent surface of
44. Conformal representation of a surface
upon
itself.
We return
and remark that another consequence of equations (83) is that the minimal curves correspond upon S and S Conversely, when two surfaces are referred to a corresponding system of lines, if the minimal lines on the two surfaces correspond, equations (83) must hold. Hence
to the consideration of conformal representation,
r
.
:
necessary and sufficient condition that the representation of two surfaces upon one another be conformal is that the minimal lines
correspond.
If the
A
minimal
lines
upon the two surfaces
ds
2
are
known and taken
as parametric, the linear elements are of the
(85)
form
ds
2
= X dadfr
= \ da^dftv
102
LINEAR ELEMENT OF A SURFACE
a conformal representation the equations
l
Hence
is
defined in the most general
way by
W
= F(a),
ft
= *;(),
or
(87)
^ = F(ft),
F
and
ft
= *;(),
which must be conjugate
where
F
1
are arbitrary functions
real.
imaginary when the surfaces are
faces referred to their minimal lines,
Instead of interpreting (85) as the linear elements of two sur we can look upon them as
lines.
the linear element of the same surface in terms of two sets of
parameters referring to the minimal
From
this point of
(86) (87) define the most general conformal of a surface upon itself. If we limit our considera representation tion to real surfaces and put, as before,
view equations
and
a
= $ + i^,
i/r
=
&lt;ty,
a1
=&lt;^ 1
+i&gt; 1 ,
ft=0 *^
1
1,
the functions fa
and
fa,
^ are
&gt;
pairs of isothermic parameters.
Now
(88)
equations (86), (87)
&lt;#&gt;
may
1
be written
1+i
= 7^^).
:
Consequently we have the theorem
When
and
a pair of isothermic parameters fa ty of a surface are
known
the
the surface is referred to the lines
=
&lt;j&gt;
const.,
= const., ^r
most general conformal representation of the surface upon obtained by making a point (fa \fr) correspond to the point
into
itself is
(fa, i^),
which
it
can be transformed in accordance with equation (88).
As
a corollary of this theorem,
we have
:
When
a pair of isothermic parameters
is
known for each of two
surfaces, all the
conformal representations of one surface upon the
directly.
ty
other can be
found
Consider two pairs of isothermic parameters fa
a surface S,
(89)
If
and
fa,
^
for
and suppose
their relation
is
&+*+! = F(t + i+).
l
two curves C and C
are in correspondence in this representa
tion, their
parametric equations must be the same functional rela
tion
between the parameters, namely,
*,)
=0.
CONFORMAL REPRESENTATION
Denote by 9 and
curves
l
103
the
the
the angles which
C and
C^
^ = const,
and
^ = const,
1
respectively.
If
make with we write
linear element of
S
in the
two forms
it
follows from (23) that
cos
a
= =
deb y
=
.
,
sin
a
=
cos
we
i
c?i/r
=,
sin
=
From
these expressions
derive the following
^
=
d(f&gt;
so that in consequence of (89)
we have
(90)
..,= :*^
7^ is the
where
function conjugate to 7^, and the accents indicate with respect to the argument. If T and F x are differentiation another pair of corresponding curves, and their angles are denoted
by 6 and
V
it
follows from (90) that
,,
OI&gt;
For, the righthand member of (90) is merely a function of the position of the point and is independent of directions. Hence in
any conformal representation defined by an equation of the form (89) the angles between corresponding curves have the same sense.
When, now,
the correspondence satisfies the equation
the equation analogous to (90)
is
Hence
l
0 =00
l
i
consequently the corresponding angles are equal in the inverse sense.
104
LINEAB ELEMENT OF A SURFACE
For the plane the
be stated thus
45. Conformal representation of the plane.
preceding theorem
may
:
The most general real conformal representation of the plane upon to the point itself is obtained by making a point (x, y) correspond where x^iy^ is any function of x + iy or x iy. (x^ y^),
We
0)
recall the
example of
41,
namely
Xl
+
iyi
=
;rrfc
is
where
c is
a real constant.
This equation
equivalent to
and also
to
C2X
Hence the
in the
on
const, and y const., in the xyplane, are represented parallels x circles which pass through the origin and have their centers the respective axes. Conversely, these circles in the xyplane correspond to
=
z^plane by
the parallels in the Xi^/iplane.
If
we put
o;2
+
y*
=
r2
,
x*
+
y*
=
rf,
equations
(ii)
and
(iii)
may
be written
&lt;*&gt;
??
fS.
Hence corresponding points are on the same line through the origin, and their On this account equations (iv) are distances from it are such that rr\ = c 2 2 2 2 2 said to define an inversion with respect to the circle x + y = c or, since TI = c /r,
,
o transformation by reciprocal radii
vectores.
From
44
it
follows that corresponding angles are equal in the inverse sense.
For the case
(v)
xi
+
iy\
=
x
c2
+
iy
the equations analogous to
(iv)
are
r
= ?, n
V
r
= Vl.
ri*
line which is the (x, y) lies on the y\) corresponding to c 2 /r. Evidently reflection in the xaxis of the line OP, and at the distance OPi
Hence the point PI (xi,
P
=
this transformation is the
combination of an inversion and the transformation
*i
=
*, y\
=
y(i)
One
finds that the transformations
line is
and
(v)
have the following properties
:
Every straight and conversely. Every
circle
transformed into a
circle
which passes through the origin ;
is
which does not pass through the origin
transformed into a
circle.
CONFOKMAL REPRESENTATION
105
propose now the problem of finding the most general conformal transformation of the plane into itself, which changes
circles not
We
passing through the origin into
circles.
#,
fi,
In solving
it
we
refer the plane to
symmetric parameters
where
a
=x
f
iy,
f$x
=
iy.
The equation
(91)
&lt;z,
of
origin is of the
any form
circle
which does not pass through the
ca(S+ aa
+ 5/3 +
;
d
0,
when the circle is real a and where 5, c, d are constants must be conjugate imaginaries and c real. Equation (91) defines
as a function of a.
If
b
@
we
differentiate the equation three times
with respect to
equations,
(92)
#,
and eliminate the constants from the resulting
2
3/3"
we
find
2/3 /3 "=0,
differentiation
where
it is
the
accent
indicates
with respect to
a.
Moreover, as equation (91) contains three independent constants,
We
(93)
the general integral of (92). know that the most general conformal representation of
the plane
upon
itself is
given by
ft
a 1 = A(a),
=
(),
or
(94)
!
=
(),
13,
= A (a).
Our problem
reduces, therefore, to the determination of functions
A
and B, such that the equation
3 ft
2
(95)
 2 ft
ft"
=0,
to
where the accent indicates differentiation with respect be transformed by (93) or (94) into (92). We consider first equations (93), which we write
a v can
Now
ff^*!L*pta_ * 30 da da
In like manner we find ft and
stituted in (95)
ft".
we
get, since A(
2
and
")
When their values are sub B are different from zero,
4
3
2
ft"
2
ffff"
+B
(3
B"
2BB
+
A
(3
l
A
2
2
A[Al")
ft
= 0.
106
LINEAR ELEMENT OF A SURFACE
it
Since equation (95) must be directly transformable into (92), follows that
(96)
3
"
2
2
"
/&gt;"
=
0,
3
A^~ 2A[A = 0.
As
these equations are of the form (92), their general integrals are similar to (91). Hence the most general forms of (93) for
our problem are
&gt;
Moreover, when, these values are substituted in an equation in a^ (S l of the form (91), the resulting equation in a and ft is of
this form.
Equation (91) may likewise be looked upon as defining a in terms of ft, so that a, as a function of ft, satisfies an equation of the form (92) similarly for a l as a function of ft r Hence if we had used (94), we should have been brought to results analogous to (97) and therefore the most general forms of (94) for our problem are
;
;
(98)
i=!4 T b /3+b^
s
:
ft 54 **,
+
,
Hence
When a plane
general conformal
circles
is
defined in symmetric parameters a,
ft,
the
most
representation of the
to circles
plane upon
itself,
for which
correspond
or straight lines, is given by (97) or (98).*
EXAMPLES
1.
Deduce the equations which
define the most general conformal representation
cZs
of a surface with the linear element
2.
2
=
dv?
+ (a2
z
z
u^dv 2 upon
itself.
Show
that the surfaces
x x
u cos v,
u cos v,
in
y
=
u
sin u,
y
u sin v,
= =
au,
a cosh * 
,
which a plane through the zaxis cuts the latter are applicable. Find the curve and deduce the equations of the conformal representation of these surfaces surface, on the plane.
3. When the representation is defined by (97), what are the coordinates of the center and radius of the circle in the &lt;nplane which corresponds to the circle of
center
(c,
d)
and radius
r in the orplane ?
* The transformations (97) and (98) play an important role in the theory of functions. For a more detailed study of them the reader is referred to the treatises of Picard, Darboux, and Forsyth.
SURFACES OF REVOLUTION
4.
107
Show
a
ai
distinct points,
that in the conformal representation (97) there are, in general, two each of which corresponds to itself also that if 7 and 5 are the
;
values of
at these points, then
d
a
K=
5
ai
+
ai
+ V(ai
a4 ) 2
+
4 a2 a3
5 Find the condition that the origin be the only point which corresponds to itself, and show that if the quantities 01, ag, ^3, a are real, a circle in the aplane through and touching the other the origin corresponds to a circle in the a^plane through
.
circle
;
also that a circle touching the xaxis at
6.
The equation 2 ai =
(a
b)
a
f


corresponds to
itself.
?
where a and
6 are constants, defines a
conformal representation of the plane upon
itself,
such that circles about the origin
and straight lines through the latter and hyperbolas in the aiplane.
in the aplane correspond to confocal ellipses
= logo: to lines parallel to the x and 7. In the conformal representation i yaxes in the aiplane there correspond lines through the origin and circles about it in the aplane, and to any orthogonal system of straight lines in the aiplane an orthogonal system of logarithmic spirals in the aplane.
is
By definition a surface of revolution the surface generated by a plane curve when the plane of the curve is made to rotate about a line in the plane. The various
46. Surfaces of revolution.
positions of the curve are called the meridians of the surface, and the circles described by each point of the curve in the revolution
are called the parallels. and for o&gt;axis and ?/axis
We
take the axis of rotation for the 2axis,
and to the
zaxis,
any two lines^perpendicular to one another, and meeting it in the same point. For any posi
tion of the plane the equation of the curve may be written z Avhere r denotes the distance of a point of the curve from the 2axis.
=
&lt;/&gt;(r),
denote the angle which the plane, in any of its positions, makes with the #2plane. Hence the equations of the surface are
let v
We
(99)
x
= r cos,v,
is
2
y
= rsinv,
2
z=(f&gt;(r).
The
linear element
(100)
If
ds
= [1 +
&lt;
(r)]
dr 2 +
we put
)
a 01
the linear element
(102)
is
transformed into
108 where X
LINEAR ELEMENT OF A SURFACE
is a function of u, which shows that the meridians and form an isothermal system. As this parallels change of parameters does not change the parametric lines, the equations
x
= u,
y
= v,
correspond
define a conformal representation of the surface of revolution upon the plane in which the meridians and to the parallels
straight lines x = const, and y = const, respectively. By definition a loxodromic curve on a surface of revolution
is
a
curve which cuts the meridians under constant angle. Evidently it is represented on the plane by a straight line. Hence loxodromic curves on a surface of revolution (99) are given by
C Vl + $*
where
a,
,
+
bv
+c=
0,
c are
constants.
Incidentally
we have
the theorem
:
When
the linear element of
a surface
is
reducible to the form
where \
is
a function of u or v alone, the surface
is
applicable
to
a
surface of revolution.
For, suppose that X
is
a function of
u
alone.
Put
r
= Vx
and
solve this equation for u as a function of r. If the resulting expression be substituted in (101), we find, bya quadrature, the function for which equations define the surface of y
&lt;f&gt;(r)
(99)
revolution with the given linear element. When, in particular, the surface of revolution
r,
is
the unit sphere,
with center at the origin, we have
r
=
sin w,
z= Vl
r
2
=
cos w,
where u is the angle which the radius vector of the point makes with the positive zaxis. Now
= log tan
Hence the equations
of correspondence are
.

x
u = log tan,
,
y=
v.
MERCATOR REPRESENTATION
109
This representation is called a Mercator chart of the sphere upon the plane. It is used in making maps of the earth for mariners. path represented by a straight line on the chart cuts the meridians
A
at constant angle.
Conformal representations of the sphere. We have found ( 35) that when the unit sphere, with center at the origin, is referred to minimal lines, its equations are
47.
a
(103)
"
+ /3
()
is
a/3l
where a and
equation of
j3
any
are conjugate imaginary. real circle on the sphere
Hence the parametric
of the
form
ca{3+aa
where a and
b
+ b/3+d=Q,
From
this it follows that the
are conjugate imaginary and c and d are real. problem of finding any conformal
representation of the sphere
upon the plane with circles of the former in correspondence with circles or straight lines of the latter, is the same problem analytically as the determination of
this
kind of representation of the plane upon
45,
it
itself.
Hence, from
the results of
follows that
All conformal representations of the sphere (103) upon a plane, with circles of the former corresponding to circles or straight lines of the latter, are defined by
a.a
&lt;
+
a,,
104)
***;{?+
^ = ^A
.
y
bfi+b.
*
We
is
wish to consider in particular the case in which the sphere represented on the ^plane in such a way that the great cir
cle
determined by this plane corresponds with
(103)
itself
point for
point.
From
we have
that the equations of this circle are
* The representation with the lower signs is the combination of the one with the upper which from (103) is seen to transform a figure sign and the transformation &i /3, /Si= bn the sphere into the figure symmetrical with respect to the zzplane.
=
,
110
LINEAR ELEMENT OF A SURFACE
these values are substituted in (104)
a
it
i
When
is
found that we
must have
i=4
,
bl
=t&gt;
V
az
=&lt;*
3
=0 =0
r
Z
3
A =i),
so that the particular form of (104)*
is
equivalent to
*1
=(+/9),
y,
=
we
()
find
From
these equations and (103)
of the straight lines joining corresponding points are reducible to and
that the equations on the sphere
plane
X
For
all
Y
1Z
Hence
values of a and ft these lines pass through the point (0, 0, 1). a point of the plane corresponding to a given point upon the point of intersection with the plane of the line the sphere is
P
with the pole (0, 0, 1). This form of representation is joining called the stereographic projection of the sphere upon the plane. It is evident that a line in the plane corresponds to a circle on
this circle the sphere the given line.
;
P
is
determined by the plane of the pole and
will close this chapter with a few remarks about the conformal representation of the sphere upon itself. From the fore
We
such representation of the going results we know that every of similar form in a^ ft v where sphere (103) is given by equations the latter are given by (86) or (87), and that for conformal repre have the values sentations with circles in correspondence a l and
^
(97) or (98).
We
consider in particular the case
a.a
of the sphere are found expressions of the linear elements to be reducible to
The
4 dad/3
2
~
4 da^ft,
4 dad ft
*
Here we have used the upper signs
in (104).
STEREOGRAPHIC PROJECTION
111
define an isometric representation of the Hence, equations (105) are preserved in the same sense sphere upon itself. Since angles this representation may be looked upon as determining
by
(105),
a motion of configuration
it.
positions if there stationary points in the general motion, upon are any, correspond to values of a and /3, which are roots of the respective equations
upon the sphere
into
new
The
If
t
l
and
1/
2
.
and
2
are the roots of the former, those of the latter are l/^ Hence there are four points stationary in the motion;
their curvilinear coordinates are
1 L\
/.
*
X
/
j
"
\
I
J.
Ln i
"
~
From
(103)
it is
seen that the
first
two are
at infinity,
and the
last two determine points on the sphere, so that the motion is a rotation about these points. If the zaxis is taken for the axis of must be oo and rotation, we have from (103) that the roots of (106) so that (105) becomes #2 hence 0, ; 3
=
=
If the rotation is real,
these equations
must be
of the
form
=
where
o&gt;
e
is
the angle of rotation.
EXAMPLES
1.
ds 2
=
2.
Find the equations of the surface of revolution with the linear element dw2 + (a2  w 2 )du 2
.
Find the loxodromic curves on the surface
X
=
MCOSU,
y
= usmv,
when
z
=
a cosh 1 ,
i
i
u
and
find the equations of the surface
referred to an orthogonal system of
these curves.
Find the general equations of the conformal representation of the oblate spheroid upon the plane.
3.
the evolute of 4. Show that for the surface generated by the revolution of to the catenary about the base of the latter the linear element is reducible 2 2 2= ds + u dv
du"
.
112
5.
LINEAR ELEMENT OF A SURFACE
A great circle on the unit sphere cuts
Find the equation of
its
the meridian v
=
in latitude
&lt;x
under
angle a.
6.
stereographic projection.
y
=
Determine the stereographic projection of the curve x acos 2 w, z asinw from the pole (0, a, 0).
=
asinwcosw,
GENERAL EXAMPLES
1.
When
there
is
crossratio of four tangents to one surface at a point the corresponding tangents to the other.
2.
a onetoone point correspondence between two surfaces, the is equal to the crossratio of
Given the paraboloid
x
=
2awcosu,
y=2&Msinv,
z
=
2
w 2 (a cos2 u
+
6sin 2 u),
where a and
b are constants.
Determine the equation of the curves on the surface,
such that the tangent planes along a curve make a constant angle with the xyplane. Show that the generators of the developable 2, enveloped by these planes, make a constant angle with the zaxis, and express the coordinates of the edge of regression
in terms of v.
Find the orthogonal trajectories of the generators of the surface S in Ex. 2. that they are plane curves and that their projections on the xyplane are involutes of the projection of the edge of regression.
3.
Show
4. Let C be a curve on a cone of revolution which cuts the generators under constant angle, and Ci the locus of the centers of curvature of C. Show that C\ lies upon a cone whose elements it cuts under constant angle.
5. When the polar developable of a curve degenerates into a point.
is
developed upon a plane, the curve
is
6.
When
the rectifying developable of a curve
line.
developed upon a plane, the
curve becomes a straight
7.
Determine
&lt;f&gt;(o)
so that the right conoid,
x
=
ucosv,
y=usinv,
z
=
(f&gt;(v),
shall be applicable to a surface of revolution.
Determine the equations of a conformal representation of the plane upon which the parallels to the axes in the aiplane correspond to lines through a point (a, b) and circles concentric about it in the aplane.
8.
itself for
9.
The equation a\
=
c sin a,
where
sentation of the plane
upon
itself
c is a constant, defines a conformal repre such that the lines parallel to the axes in the
aplane correspond to confocal ellipses and hyperbolas in the aiplane.
10. In the
conformal representation of the plane upon
itself,
given by ai
= a2
,
to lines parallel to the axes in the oriplane there correspond equilateral hyperbolas in the aplane, and to the pencil of rays through a point in the oriplane and the cir
about it there corresponds a system of equilateral hyperbolas through the corresponding point in the orplane and a family of confocal Cassini ovals.
cles concentric
11.
When
curves, the
12.
sum
the sides of a triangle upon a surface of revolution are loxodromic of the three angles is equal to two right angles.
of a sphere
The only conformal perspective representation
upon a plane
is
given by (104).
GENERAL EXAMPLES
13.
113
Show
interchange of
that equations (105) and the equations obtained from (105) by the cc. and /3 define the most general isometric representation of the
sphere upon
14.
itself.
Let each of two surfaces S, S\ be defined in terms of parameters w, u, and points on each with the same values of the parameters correspond. If H\, for S, corresponding elements where the latter is the function for Si analogous to
let
H
H
of area are equal and the representation is said to be equivalent.* If the parameters of S are changed in accordance with the equations
the condition that the equations u resentation of S and Si is
v a),
H ^ HI and
&lt;f&gt;
u
(w, v),
= $ (u,
= M,
v
=v
define
an equivalent rep
H
\[&lt;)
du dv
15.
cv du
HI (0,
Under what conditions do
x
aix
the equations
+
azy
+
a3
,
y
= b& +
upon
b2 y
+
63
define an equivalent representation of the plane
16.
itself ?
Show
that the equations
determine an equivalent representation of the surface of revolution (99) upon the
plane.
17. Given a sphere and circumscribed circular cylinder. If the points at which a perpendicular to the axis of the latter meets the two surfaces correspond, the
representation
is
equivalent.
18. Find an equivalent representation of the sphere upon the plane such that the parallel circles correspond to lines parallel to the yaxis and the meridians to a, 0). ellipses for which the extremities of one of the principal axes are (a, 0), (
*
German
writers call
"
it
flachentreu."
CHAPTER IV
GEOMETRY OF A SURFACE
48.
IN
THE NEIGHBORHOOD OF A POINT
we study
of
it,
Fundamental coefficients of the second order. In this chapter the form of a surface in the neighborhood of a point and the character of the curves which lie upon the surface
M
and pass through the point.
all
We
recall that the tangents at
the tangent
M to
these curves
lie in
a plane,
plane to the surface at
the point.
The equation may be written
(1)
of the tangent plane at
M(x,
y, 2),
namely
(II, 11),
(f
where we have put
dz
_ _
dx du
dx
dv
dx
dti
du
du
dz
du
dy du
dy dv
H
We
for
dy^
~dv
H
dz
H
dx
dv
dv
do
which the functions X,
it
define the positive direction of the normal ( 25) to be that 7 I Z are the directioncosines. From this
,
const, and follows that the tangents to the curves v = const, at a point and the normal at the point have the same u mutual orientation as the #, ?/, and 2axes.
definition
From
(3)
(2)
follow the identities
F ^ 7T
dv
fact that the
 fl u
which express the
normal
is
perpendicular to the tan
gents to the coordinate curves. In consequence of these identities the expression for the distance p from a point du, v + dv) (u is of the second order in du and dv. to the tangent plane at
M
+
M
It
(4)
may
be written
p=
^X dx = 1 (D du +2D dudv +
2
&"
dv
2
)
+ e,
lit
COEFFICIENTS OF THE SECOND OKDEK
where
e
115
denotes the aggregate of terms of the third and higher are defined by orders in du arid dv, and the functions D\
D"
Z&gt;,
(5)
dudv
equations
(3)
If
be differentiated with respect to u and v respec
tively,
we
get
dv du
(6)
1
=
dX
0,
dx
_
~
dudv
dv
dv dv
And
so equations (5)
may
be written
(7)
_V Y ~^^
,,_y
The quadratic
dudv~
Ztdu
d
dv du
** 9*
^__y^^ ^
dv dv
differential
&lt;1&gt;
form
(8)
is
= D du* + 2 D
dudv
2
f
D"dv
called the second fundamental form of the surface,
and the func
tions D,
D
,
D"
the fundamental
coefficients
of the second order.
We
leave
first
it
to the reader to
show that these
coefficients, like those of the
order, are invariant for
will
we shall now be derived. From the equations
Later
any displacement of the surface in space. have occasion to use two sets of formulas which
of definition,
(9)
^ \du]
get,
^ du dv
~ toy
I
cv)
:
we
by differentiation and simple reduction, the following
a^^_ia^
dx
(10)
y^^ = ?^_l?^,
v dx
&x
_l
dE
dG
du
dv dudv
~2
2 dv
116
Again,
GEOMETRY OF A SURFACE ABOUT A POINT
if the expressions (9) be substituted in the lefthand mem bers of the following equations, the reduced results may be written by means of (2) in the form indicated
:
"
dv
du
\
du
du,
dv
du
\
dv
dv,
x, y, z
;
Similar identities can be found by permuting the letters
X, F, Z.
From
the fundamental relation
we
obtain,
by differentiation with respect
to
u and
v respectively,
the identities
These equations and
linear in
*
(7)
constitute a system of three equations
du
ij.
du
O y
du
and a system linear in
find,
dv
,
,
dv
dv
Solving
y
for
and for
du
dv
we
by means of
(11),
dX
~du~
"
FD GDdx
H*
du
dx
FDED
H*
FZ&gt;
dx
^
dv
^dX _FD"GD
dv
7}
RD"dx
H*
du
H*
dv
V
The
and
expressions for
..
,
du
,,
7 /}
are obtained
dv
by replacing x by y
z respectively.
first
By means of these equations we shall prove that a real surface whose second fundamental coefficients are in proportion, thus
(14)
and
D=V=
V"
=\
We
assume
D"
where X denotes the factor of proportionality, is a sphere or a plane. that the minimal lines are parametric. In consequence we have
E=G=D=
so that equations (13)
=
0,
become
dX \
du du
dv
\ dv
RADIUS OF NORMAL CURVATURE
The function X must
satisfy the condition
117
dv \
du/
0.
du
\
dv
which
reduces to
du du
dv
=
dv
Moreover, we have two other equations of
z respectively.
condition, obtained from the above the proportion
by replacing x by y and
.
Since
to.ay to
= to
dv
.0y
dv
to
.
du du du
is
dv
:
== that is, X is a condu dv stant. When X is zero the functions X, F, Z given by (15) are constant, and consequently the surface is a plane. When X is any other constant, we get,
not possible for a real surface,
we must have
by integration from
(15),
X
where
\x
+
a,
Y=
\y
4 6,
Z = Xz +
we
c,
obtain (\x f a) 2 4 (\y 4 6) 2 2 1. Since this is the general equation of a sphere, it follows that the 4 (Xz 4 c) above condition is necessary as well as sufficient.
a, 6, c are constants.
From
these equations
v/
49.
Radius of normal curvature.
Consider on a surface
S any
a point M. The direction of its tangent, MT, Let denote the angle which is determined by a value of dv/du. the positive direction of the normal to the surface makes with the
curve
C through
o&gt;
positive direction of the principal normal to C at Jf, angles being measured toward the positive binormal. If we use the notation of the first chapter, and take the arc of C for its parameter, we have
In terms of
the forms
,,
.
and
as as
the derivatives in the parenthesis have
fo = Zfa/duV ~~
aV
du \ds)
is
2
2
du dv ds ds
equivalent to
dv \ds
2
so that the above equation
cos
(16)
w
D du + 2 D dudv +
2
D"dv
Edu*+ ZFdudv 4 Gdv*
As the righthand member of this equation depends only upon the curvilinear coordinates of the point and the direction of MT,
it is
the same for all curves with this tangent at M. Since p is cannot be greater than a right angle for one positive, the angle
o&gt;
curve tangent to
MT,
if it is less
than a right angle for any other
118
GEOMETRY OF A SURFACE ABOUT A POINT
MT
curve tangent to MT; and vice versa. We consider in particular the curve in which the surface is cut by the plane determined by
and the normal
tangent to
to the surface at
M.
We
call it the
normal
the
the
MT, and let p n denote its radius. Since member of equation (16) is the same for C and righthand normal section tangent to it, we have
section
(17)
P
Pn
is less or +1 1, according as greater than a right angle; for p and p n are positive. Equation (17) gives the follow
where
e is
or
w
ing theorem of Meusnier:
The center of curvature of any curve upon a surface
jection
is
the
pro
osculating plane of the center of curvature of the normal section tangent to the curve at the point.
upon
its
In order to avoid the ambiguous sign in (17), we introduce a
new function when
R which
is
equal to pn
when
&lt;
o&gt;
&lt;
TT/%,
and to
pn
7r/Z&lt;a&gt;&lt;7r,
and
call it the radius of normal curvature of the
surface for the given direction
MT. As
thus defined,
E is given
by
R
Now we may
state
Edu + 2 Fdudv + Gdv*
2
Meusnier
s
theorem as follows
:
If a segment, equal to twice the radius of normal curvature for a given direction at a point on a surface, be laid off from the point on the normal to the surface, and a sphere be described with the segment
for diameter, the
circle in ivliich the sphere is met by the osculating a curve with the given direction at the point is the circle of plane of curvature of the curve.
50. Principal radii of
normal curvature.
If
we put
t
=
&gt;
equa
tion (18) becomes
I
D+2D
t+D"t
When
of
t,
the proportion (14) is satisfied, R is the same for all values 1/X for the sphere. being oc for the plane, and the constant
For any other surface
R
varies continuously with
t.
And
so
we
PRINCIPAL RADII OF CURVATURE
seek the values of
119
To
to
t
this
t for which 11 is a maximum or minimum. end we differentiate the above expression with respect and pnt the result equal to zero. This gives
2
(20)
(J}
+D"t)(E+2Ft+Gt
l
)(F+Gt)(D + 2D t +
2
D"t
)
= Q,
or
2
(21)
(FD"GD )t
+(FD"GD)t+(ED
FD) =
().
Without any
loss of generality
curves are such that
(22)
(ED"
E
=
0,
so that
r
we can assume that the parametric we have the identity
H
J
2
GDf 4
2
7T"
(FD"
D G) (ED FD)
FD)\.
\_
=4
When
is
E
T 2F (FD FDf+\ ED"GD(ED E
is real,
the surface
and the parameters
is
member
of this equation
positive.
also, the righthand Since the lefthand member
the discriminant of equation (21), the latter has two real and distinct roots.* When the test (III, 34) is applied to equation (21), it is found that the two directions at a point determined by the
roots of (21) are perpendicular.
Hence:
At
which
the radius of
it is
every ordinary point of a surface there is a direction for which normal curvature is a maximum and a direction for
a minimum, and they are at right angles
to
one another.
These limiting values of R are called the principal radii of normal curvature at the point. They are equal to each other for the plane and the sphere, and these are the only real surfaces
with this property.
From
(20)
and
(19)
we have
D
+D"t_D
F+Gt
:
+D _ 1 E + Ft~~R
t
Hence the following relations hold between the principal radii and the corresponding values of t
E + FtR(D + D = Q, = 0. \F+GtR(D +
f
t)
D"t)
* In order that the two roots he equal, the discriminant must vanish. This is impos sible for real surfaces other than spheres and planes, as seen from (22). For an imaginary surface of this kind referred to its lines of length zero, we have from (21) that or D"
D
is
zero, since
F ^ 0.
The vanishing
condition that the numerator and
of the discriminant is also the necessary and sufficient denominator in (19) have a common factor.
120
GEOMETRY OF A SURFACE ABOUT A POINT
t is
When
(24)
eliminated from these equations,
we
get the equation
2 )
(DD"
D A 1
2
)2
)
(ED"
+ GD2 FD R + (EG  F = 0,
)
whose roots are the principal and /3 2 we have
,
radii.
If these roots
be denoted by p l
!_
^^
(25)
DD"D
2
PiP*
H
and a
it
Although equations plane, and for no other
(14) hold at all points of a sphere
surface,
may happen
that for certain par
ticular points of a surface they are satisfied. At such points R, as given by (19), is the same for all directions, and the equa
tion (21) vanishes identically. When points of this kind exist they are called umbilical points of the surface.
EXAMPLES
1.
When
the equation of the surface
is z
=f(x,
?/),
show that
x,Y,z =
D,
^J^,
2z
D
dz
&gt;
,
D"
=
where
2.
p
Show
=
dz
&gt;
a
r
=8
d 2z
s
dx
dy
dx*
dxdy
that the normals to the right conoid
along a generator form a hyperbolic paraboloid.
3.
Show
that the principal radii of normal curvature of a right conoid at a
point differ in sign.
4.
tion at a point in the direction of the loxodromic curve through
Find the expression for the radius of normal curvature of a surface of revolu makes the it, which
the meridians.
angle
5.
a with
Show
sin
i&gt;,
y
=u
that the meridians and parallels on a surf ace of revolution, x = u cosu, in which the radius of normal curvature is z= (w), are the directions
;
maximum and minimum
Pl
that the principal radii are given by
(1
+
/2
)
P2
M
and that
the segment of the normal between the point of the surface and the intersection of the normal with the zaxis.
/&gt;
2 is
6.
Show
that AIX
=1
X
2
eters are
formed with respect
and AI (x, y) XY, where the differential to the linear element of the surface.
=
param
LINES OF CURVATUBE
51. Lines of curvature.
121
Equations of Rodrigues.
We have
f
seen
that the curves defined
(26)
by equation (21), written
(ED"
(ED
 FD) du +
2
 GD) dudv +
(FD"
 GD
)
dv
z
=0,
form an orthogonal system. As defined, the two curves of the sys tem through a point on the surface determine the directions at the
point for which the radii of normal curvature have their and minimum values. These curves are called
maximum
the lines of curvature, and their tangents at a point the principal directions for the point.
They possess another geometric property which
we shall now find. The normals to
face
form a ruled surface.
a surface along a curve In order that the sur
be developable, the normals must be tangent to a curve ( 27), as in fig. 12. If
the coordinates of a point on the normal l at a point be denoted by x r y^ z^ we have
M
M
FIG. 12
where
r denotes the length
MMr
If
M^ be
a point of the edge of
regression,
we must have
dx+rdX+Xdr _dy{r dY+ Ydr _dz + r dZ + Zdr X Z Y
tive
Multiplying the numerators and the denominators of the respec members by X, F, Z, and combining, we find that the common ratio is dr. Hence the above equations reduce to
or,
when
the parametric coordinates are used,
(
8x du
du
,
dx
H
dv
,
dv
+r
dX
I
du
,
,
dX
dv
H
dv
,
\du
(27)
dv
fa
du
dv
dv
du
dv
122
GEOMETRY OF A SURFACE ABOUT A POINT
be multiplied by
dz dv
&gt;
If these equations
added, and by
dx
dv
^u
^u
^u
respectively and
o
^ dv
respectively and added,
we
get
Fdu + Gdv
But
The normals
to
r(D du
+
D"
dv)
= 0.
these equations are the same as (23).
Hence:
a surface along a curve of it form a ruled surface which is a developable only when the curve is a line of curvature ; in this case the points of the edge of regression are the centers of normal
curvature of the surface in the direction of the curve.
The
(28)
coordinates of the principal centers of curvature are
the parametric curves are the lines of curvature, equa tion (26) is necessarily of the form
(29)
When
X dudv
=
0,
and consequently we must have
Since ED"GD
(30)
=
ED
FD = 0,
=Q.
FD"
GD = 0.
0,
these equations are equivalent to
^=0,
D
Conversely, when these conditions are satisfied equation reduces to the form (29). Hence:
(26)
A
curvature be necessary and sufficient condition that the lines of
is
parametric
that
F and D
be zero.
Let the
the principal tions of the lines of curvature v
lines of curvature be parametric, and let p^ and p 2 denote radii of normal curvature of the surface in the direc
= const, and u = const, respectively.
From
(31)
(19)
we
find
^
(13)
=
f
By du
~=
^T
dz
and equations
du
(32)
become
dY
ri
_
dZ
du
du
du
These equations are called the equations of Rodrigues.
TOTAL AND MEAN CURVATURE
52. Total
123
in
and mean curvature.
Of fundamental importance
the discussion of the nature of a surface in the neighborhood of a point are the product and the sum of the principal curvatures at
the point. They are called the total curvature * of the surface at the point and the mean curvature respectively. If they be denoted
by
K and K
m1
we
have, from (25),
K
(33)
1
Mb"
^
JLi+iPi P*
^
two principal
radii
When
K
is
positive at a point J/, the
have
the same sign, and consequently the two centers of principal curva ture lie on the same side of the tangent plane. As all the centers of curvature of other normal sections lie between these two, the
lies entirely on portion of the surface in the neighborhood of one side of the tangent plane. This can be seen also in another
M
way.
Since
H
2
is
positive,
we must have
DD
f
D
2
&gt;
0.
Hence
the distance from a nearby point to the tangent plane at Jf, since it is proportional to the fundamental form ( 48), does not
&lt;l&gt;
change sign as dv/du
is
varied.
negative at M, the principal radii differ in sign, and consequently part of the surface lies on one side of the tangent plane and part on the other. In particular there are two directions,
is
When
K
given by
for
j&gt;du*+2
D dudv +
is
D"
dv 2 =
0,
which the normal curvature
zero.
In these directions the dis
tances of the nearby points of the surface from the tangent plane, as given by (4), are quantities of the third order at least. Hence
these lines are the tangents at meets the surface. plane at
M
M to the curve in which the tangent
zero,
At
infinite.
the points for which At these points
K
&lt;
is
one of the principal radii
f
is
has the form (^/J) du
vanishes in the direction
dv/du passes does not change sign. through the value given by this equation, Hence the surface lies on one side of the tangent plane and is tan gent to it along the above direction.
&lt;1&gt;
Vl)du
+ ^/D"dv =
^W dvf
and
0.
But
as
* The total curvature is sometimes called the Gaussian curvature, after the celebrated geometer who suggested it as a suitable measure of the curvature at a point. Cf Gauss, p. 15.
.
124
GEOMETEY OF A SUEFACE ABOUT A POINT
may
anchor ring, or tore, is a surface with points of all three kinds. Such a sur be generated by the rotation of a circle of radius a about an axis in the plane of the circle and at a distance b a) from the center of the circle. The points at the distance b from the axis lie in two circles, and the tangent plane to the tore at a point of either of the circles is tangent all along the circle. Hence the surface has zero curvature at all points of these circles. At every point whose dis tance from the axis is greater than b the surface lies on one side of the tangent
face
(&gt;
An
plane, whereas,
when
the distance
is less
than
6,
the tangent plane cuts the surface.
There are surfaces for which is positive at every point, as, for example, the ellipsoid and the elliptic paraboloid. Moreover, for the hyperboloid of one sheet and the hyperbolic paraboloid the
curvature
is negative at every point. Surfaces of the former type are called surfaces of positive curvature, of the latter type surfaces
K
of negative curvature.
is zero at 64) we shall prove that when a surface the latter is developable, and conversely.
Later
(
K
all
points of
Equation of Euler. Dupin indicatrix. When the lines of curvature are parametric, equation (18) can be written, in con
53.
sequence of
(34)
(III,
23) and (31), in the form
2 cos #
I
sin
2
6&gt;
Pi
Pi
where
the angle between the directions whose radii of normal curvature are and p r Equation (34) is called the equation of Euler.
is
R
When
is
the total curvature
K at a point
and p 2 for the point have the same sign, and R has this sign for all
positive, p l
directions.
If the
of curvature at the point
FlG 13

tangents to the lines be taken for
M
coordinate axes, with respect to which %
and
T?
are coordinates,
and segments of
is
length
VTIFi be laid off from
M in the two directions correspond
the ellipse
1.
ing to R, the locus of the
(fig.
end points of these segments
13) whose equation
is
N + ra =
Dupin
&
This
ellipse
is
called the
indicatrix for the point.
particular, p l
Dupin
and p 2 are equal, the indicatrix is a circle. indicatrix at an umbilical point is a circle ( 50).
is
When, in Hence the
For
this
reason such a point
sometimes called a circular point.
DUPIN INDICATRIX
When
125
negative p l and p 2 differ in sign, and consequently certain values of R are positive and the others are negative. In the directions for which R is positive we lay off the segments
is
K
V.Z2,
and
in the
other directions
V
R.
The
locus of the
end points of these segments con sists of the conjugate hyperbolas
(fig.
14) whose equations are
T]_
Pl
Pt
We
(35)
or,
remark that
R
is infinite
for
the directions given by
tah
2
0=^,
directions of the
is
FIG. 14
in other words, in the
asymptotes to the
hyperbolas.
Finally,
The above when K =
locus
the
Dupin
the equation of the indicatrix
e2_
i
indicatrix for the point. is of one of
the forms
that
is,
a pair of parallel straight lines. In view of the foregoing results, a point of a surface is called elliptic, hyperbolic, or parabolic, according as the total curvature at the point is positive, negative,
or zero.
1
the expression for the distance p upon the of from a nearby point tangent plane to a surface at a point the surface is given by
In consequence of
(4)
M
P
Edu
Pl
2
+
Gdv2
=
n 2
,
to within terms of higher order.
But ^/Edu and ^/Gdv
are the
distances, to within terms of higher order, of in the directions of the lines of curva planes to the surface at
P
from the normal
M
ture.
Hence the plane
parallel to the tangent plane
and at a
dis
tance
p
from
it
cuts the surface in the curve
Evidently this
is
a conic similar to the
point,
Dupin
indicatrix at an
elliptic or parabolic
and
to a part of the indicatrix at a
hyperbolic point.
126
GEOMETRY OF A SURFACE ABOUT A POINT
EXAMPLES
1. Show that the meridians and parallels of a surface of revolution are its lines of curvature, and determine the character of the developable surfaces formed by the normals to the surface along these lines.
2.
Show
that the parametric lines on the surface
X
are straight lines.
3.
a = (ttM),
,
y
=
b
( U v),
,
z
= uv ,
Find the
is
lines of curvature.
When
a surface
denned by z
= /(x,
?/),
the expressions for the curvatures are
and the equation of the
[(1
4.
lines of curvature
[(i
is
+
p2) s
 pqr] (to* +
2
i
+ p2) t(l +
g2) r
j
dxdy
+
[
pqt
_
(1
+
?2) s]
dy z
=
.
The
principal radii of the surface y cos
o 2 _i_
x sin 
=
at a point
(x, y, z)
are
n2
equal to
5.
Find the
lines of curvature.
Derive the equations of the tore, defined in
52,
and prove therefrom the
is
results stated.
6. 7.
The sum
of the
normal curvatures
in
two orthogonal directions
constant.
The Euler equation can be written
E=
Pi
2plp *
+
P2

(PI

pa)
cos 2 6
54. Conjugate directions at a point. Conjugate systems. Two are said to have conjugate curves on a surface through a point coincide with conjugate diam directions when their tangents at
M
M
indicatrix for the point. These tangents are also parallel to conjugate diameters of the conicr in which the sur and very face is cut by a plane parallel to the tangent plane to
eters of the
Dupin
M
the point in denote a point of this conic and near it. Let which its plane a cuts the normal at M. The tangent plane to to the meets the plane a in the tangent line at the surface at
P
N
P
P
Moreover, this tangent line is parallel to the diameter conju approaches 3/this tangent line approaches gate to NP. Hence as the diameter of the indicatrix, which is conjugate to the
conic.
P
Dupin
diameter in the direction
MP. Hence we have
(cf.
27)
:
The
of
characteristic of the tangent plane to a surface, as the point contact moves along a curve, is the tangent to the surface in the
direction conjugate to the curve.
CONJUGATE DIRECTIONS
By means
of this
127
theorem we derive the analytical condition for
is
conjugate directions. If the equation of the tangent plane
f,
77,
f being current coordinates, the characteristic
is
denned by
this equation,
and
where
moves.
s is
If &c, 8y, Bz
the arc of the curve along which the point of contact denote increments of #, ?/, z in the direction
conjugate to the curve,
we
have, from the above equations,
If
Bu and
(36)
increments of u and v in the conjugate direction be denoted by 8v, this equation may be written
D duBu + D (du8v + dvSu) + D"dvv = 0.
f
The
(37)
directions conjugate to any curve of the family
&lt;(%,
v)
= const.
are given
(
by
cv
38)
du
dv
first
du
order and
first
As
it
this is a differential
equation of the
degree,
defines a oneparameter family of curves. These curves and the curves const, are said to form a conjugate system. Moreover,
=
&lt;/&gt;
any two families
of curves are said to
form a conjugate system
when the tangents
to a curve of each family at their point of inter
section have conjugate directions. From (36) it follows that the curves conjugate to the curves v = const, are defined by Sv = 0. Consequently, in order Su +
D
D
that they be the curves u = const., we As the converse also is true, we have
must have
:
D
1
equal to zero.
A necessary and sufficient condition form a conjugate system is that D be
that the parametric curves
zero.
128
GEO vlETRY OF A SURFACE ABOUT A POINT
have seen
We
(51)
that the lines of curvature are characterized
by the property that, when they are parametric, the coefficients are zero. Hence and
F
D
:
The
lines
of curvature
form a
conjugate system
and
the
only
orthogonal conjugate system.
If the lines of curvature are parametric,
and the angles which
to the curve
a pair of conjugate directions
v
= const,
are denoted
by
make with the tangent and 6 we have
, ,
tan
a
=
[G dv ^J MjE du
&lt;
ar tan 6 = x
N^ Su
[GSv 
,
so that equation (36)
(39)
may
be put in the form
tan0tan0 = ?H,
is
which
55.
the wellknown equation of conjugate directions of a conic.
is equal to 0, Asymptotic lines. Characteristic lines. When reduces to (35). Hence the asymptotic directions are equation (39) = dv/du, we obtain selfconjugate. If in equation (36) we put Sv/Su
(40)
D du + 2 D dudv +
2
D"
dv
2
= 0,
at each
which determines, consequently, the asymptotic directions
point of the surface.
This equation defines a double family of curves upon the surface, two of which pass through each point and admit as tangents the asymptotic directions at the point. They
are called the asymptotic lines of the surface. The asymptotic lines are imaginary on surfaces of positive curva ture, real on surfaces of negative curvature, and consist of a single real family on a surface of zero curvature.
Recalling the results of
52,
we
say that the tangent plane to
a surface at a point cuts the surface in asymptotic lines in the neighborhood of the point. As an immediate consequence, we have that the generators of a ruled surface form one family of
asymptotic lines. Since an asymptotic line
is
selfconjugate, the characteristics of
the tangent plane as the point of contact totic line are the tangents to the latter.
moves along an asymp Hence the osculating
plane of an asymptotic line at a point
is
the tangent plane to the
ASYMPTOTIC LINES
129
surface at the point, and consequently the asymptotic line is the edge of regression of the developable circumscribing the surface
along the asymptotic line. This follows also from equation From (40) we have the theorem
:
(16).
necessary and sufficient condition that the asymptotic lines upon a surface be parametric is that
D=D"=Q.
If these equations hold, and, furthermore, the
A
parametric curves
is
are orthogonal,
it is
seen from (33) that the
:
mean curvature
zero,
and conversely. Hence A necessary and sufficient condition
an orthogonal system
is
that the asymptotic lines
form
that the
mean curvature of
is
the surface be zero.
A
surface
whose mean curvature
a minimal surface.
sists of
At
zero at every point is called each of its points the Dupin indicatrix con
two conjugate equilateral hyperbolas. means of (39) we find that the angle between conjugate By directions is given by
P/&gt;1
consider only real lines, this angle can be zero only for sur faces of negative curvature, in which case the directions are asymp
If
totic.
we
It is natural, therefore, to
seek the conjugate directions upon
a surface of positive curvature for which the included angle is a minimum. To this end we differentiate the righthand member of
the above equation with respect to 6 and equate the result to zero. The result is reducible to
(41)
tan 6
(39)
=
:
Then from
we have
From
these equations
it
follows that #
=
0,
and
Conversely,
when
=
6 equation (39) becomes (41).
Hence:
surface of positive curvature there is a unique conjugate system for which the angle between the directions at any point is the
Upon a
130
GEOMETRY OF A SURFACE ABOUT A POINT
minimum
angle between conjugate directions at the point ; it is the only conjugate system whose directions are symmetric with respect to the directions of the lines of curvature.
These
lines are called the characteristic lines.
It is of interest
to note that equations (35)
and
(41) are similar,
and that the
real
upon a surface of negative curvature are symmetric with respect to the directions of the lines of curvature. As just seen, if 6 is the angle which one characteristic line makes
asymptotic directions
with the line of curvature
teristic line
v
= const,
6.
at a point, the other charac
makes the angle
ture for these directions are
Hence the radii of normal curva equal, and consequently a necessary
and
sufficient condition that the characteristic curves of a surface
is
be parametric
(42)
f 7T
=
iy=
By reasoning
similar
56. Corresponding
to that of
systems on two surfaces. 34 we establish the theorem
:
and sufficient condition that the curves defined by 2 fidu +2S dudv + T dv = form a conjugate system upon a surnecessary
2
A
face
*s
RD"
+ TD
:
2
SD = 0.
From
this
we have
at once
D
If the second quadratic forms of two surfaces S and S are 2 2 dv and D l du + 2 D[ dudv + D[ dv\ and if a du 2 + 2 D dudv +
l
f
D"
to the point on the other point on one surface is said to correspond u and tf, the equation with the same values of
(43)
du2 dudv
dv
2
D?
D[
DI
is
D"
D D
conjugate for both surfaces.
real
which defines a system of curves
By the methods
curvature of
of
50 we prove that these curves are
when
If the
either or both of the surfaces S,
S
l
S
is
negative and
it is
is of positive curvature. referred to its asymptotic lines,
the above equation reduces to
GEODESIC CURVATUKE
Hence the system is real when is, when the curvature of S is
l
131
D
l
and D[ have the same sign, that
positive.
Another consequence
of the above
theorem
is
:
necessary and sufficient condition that asymptotic lines on one of two surfaces $, Sl correspond to a conjugate system on the other is
(44)
A
DDJ +
D"Di
 2 D D[ = 0.
EXAMPLES
1. Find the curves on the general surface of revolution which are conjugate to the loxodromic curves which cut the meridians under the angle a.
2. Find the curves on the general right conoid, Ex. gate to the orthogonal trajectories of the generators.
3.
1, p. 56,
which are conju
When
the equations of a surface are of the form
x=U
where U\ and
l,
y=Vi,
z=U +V
2
2
,
U
2
are functions of u alone, and V\ and
F
2
of v alone, the para
metric curves are plane and form a conjugate system.
4.
Prove that the sum of normal radii at a point
in conjugate directions
is
constant.
5.
When
a surface of revolution
is
referred to
its
meridians and parallels, the
asymptotic lines can be found by quadratures.
6.
Find the asymptotic
x
lines
on the surface
y
lines
=
a(l +
cos it) cot v,
= a(l +
cosw),
z
z=
acosw
and determine
7.
Determine the asymptotic
upon the surface
3
?/
y sin a: and their orthog
onal trajectories.
8.
Show
that the xaxis belongs to one of the latter families.
their projections
9.
Find the asymptotic lines on the surface 2 on the xyplane.
 2 xyz + z 2 = 0,
Prove that the product of the normal radii in conjugate directions
is
a maxi
mum
for characteristic lines
and a minimum for
lines of curvature.
10.
When
the parametric lines are any whatever, the equation of character
istic lines is
[D(GD  ED")  2D (FD  ED )] tin* + 2 [D (GD + ED")  D"(GD  ED")] dv* = 0. + [2D (GD FD")
2
FDD"]
dudv
57. Geodesic curvature.
Geodesies.
Consider a curve
C upon
a
of C. surface and the tangent plane to the surface at a point this tangent plane the portion of the Project orthogonally upon curve in the neighborhood of M, and let C denote this projection.
1
M
The curve C
is
1
is
a curve
upon
a normal section of the projecting cylinder, and C at M. Hence the theorem the latter, tangent to
C"
132
of
GEOMETRY OF A SURFACE ABOUT A POINT
Meusnier can be applied to these two curves. If l/p g denotes the curvature of C and ^ the angle between the principal normal to C and the positive direction of the normal to the cylinder at Jf, we have
(45)
i = c ^.
P,
P
it
In order to connect this result with others,
is
necessary to
define the positive direction of the normal to the cylinder. This normal lies in the tangent plane to the surface. make the convention that the positive directions of the tangent to the curve,
We
the normal to the cylinder, and the normal to the surface shall
have the same mutual orientations as the positive
or,
y,
and
2axes.
From
this choice of direction it follows that
if,
as usual, the direc
tioncosines of the tangent to the curve be dx/ds, dy/ds, dz/ds, then those of the normal to the cylinder are
(46 )
Y*Z*/, ds ds
of
Z~*~&gt;
ds
f
ds
*f4ds
ds
is called the geodesic curvature of (7, and p g the radius of geodesic curvature. And the center of curvature of C is called the center of geodesic curvature of C.
The curvature
C
definition the geodesic curvature is positive or nega tive according as the osculating plane of C lies on one side or the
its
From
other of the normal plane to the surface through the tangent to C. From (45) it follows that the center of first curvature of C is the
projection curvature.
upon
osculating plane of the center of geodesic Moreover, the former is also the projection of the
its
center of curvature of the normal section tangent to C (49). of (7, normal to the line Hence the plane through a point
M
curvature at M, is the joining the centers of normal and geodesic intersection with the osculating plane of C for this point, and its is the center of first curvature. join
the angle which the positive direction of the normal to the surface makes with the positive
By
definition
(
49)
w denotes
direction of the principal normal to (7, angles being measured toward the binomial. Hence equation (45) can be written
(47)
1  = sin w
GEODESIC CUEVATUKE
133
These various quantities are represented in fig. 15, for which the tangent to the curve is normal to the plane of the paper, and The directed lines MP, MB, MK, is directed toward the reader.
MN represent respectively the
positive directions of the principal
normal and binomial of the curve and
the normals to the projecting cylinder and to the surface.
curve whose principal normal at every point coincides with the normal
to the surface
A
upon which
it
lies,
is
called a geodesic.
From
that a geodesic a curve whose
may
(45) it follows also be defined as
geodesic
curvature
is
zero at every point. meridians of a surface of revolution are
in geodesies, as follows from the results
For example, the
46.
A
twisted curve
is
a
geodesic on
lies
011
its
when a straight rectifying developable, and
it
line
a surface,
shall
to
make
a geodesic for the surface. Later we an extensive study of geodesies, but now we desire
is
an expression for the geodesic curvature in terms of the fundamental quantities of the surface and the equation of
find
the curve.
58.
Fundamental formulas. The directioncosines
(
of the prin
cipal normal are
8)
d 2x
H
f\ _
~
ds
2
Q ^
d 2y
ds
2
Q r
d 2z
ds
2
.
Consequently, by means of
the form
1
(
(46),
equation (45)
may
be put in
48 )
7 *\
g
= ^\ /
dz
dy\ d x
2
ds~
v,
~ds)~ds
Expressed as functions of u and
the form
the quantities j
ji
are
^
dx
ds
_ fa
du
dx dv
dv ds
d*x
2
du ds
~ds
2
_ ~
du \ds)
2
g
d
2
x
dudv
dudv ds ds
dv \ds)
^Y+ du ds
2
+
dv ds
2
134
GEOMETRY OF A SURFACE ABOUT A POINT
these expressions are substituted in (48), and in the reduction of (10) and (11), we obtain
When
we make use
ds
ds
where
~
L
and
_
M have the significance
+
.
~
~
du
ds
ds^\dv
,_ 29/Vb/
+~
2 dv \ds)
,
F
&lt;*r
(Fv
ds^
\\
^^
du ds ds
(\\ F
D
,
2 ~dv) \ds)
ds*
*
Gdepends
From
this it is seen that the geodesic curvature of a curve
D".
upon E, F, G, and is entirely independent of D, Suppose that the parametric lines form an orthogonal system, and that the radius of geodesic curvature of a curve v = const, be
denoted by p gu
.
In this case
F= 0,

ds
= Vfldu.
Hence the above
equation reduces to
(50)
r
we
is
In like manner
find that the geodesic curvature of a curve
u = const,
given by
_!
As an immediate consequence
theorem
:
c^ these equations
f
we have the
When
or
the parametric lines
upon
a surface
form an orthogonal
system, a necessary
and
that the curves v sufficient condition
is
u = const,
be geodesies
that
E be
is
a function of u alone or
= const, G of v
alone respectively.
of expressible as a function ele differential parameters of v formed with respect to the linear
It will
now
be shown that p gu
ment
(III, 4).
From the definition that when ^=0
of these parameters
(
37, 38)
it
follows
l
d
IE
~V\G
GEODESIC CURVATURE
Hence, by substitution in
(52)
(50),
135
we
obtain
JUPAL
P gn
./
1
[y&^v
In like manner, we find
(53)
i=
Thus we have shown that the geodesic curvature
line is a differential
line.
of a parametric
parameter of the curvilinear coordinate of the Since this curvature is a geometrical property of a line, it is
necessarily independent of the choice of parameters, and thus is an invariant. This was evident a priori, but we have just shown that it is an invariant of the differential parameter type.
the definition of the positive direction of the normal to a surface ( 48), and the normal to the cylinder of projection, it fol lows that the latter for a curve v = const, is the direction in which v
increases, whereas, for a curve u const., it is the direction in which u decreases. Hence, if the latter curves be defined by u = const., equations (52) and (53) have the same sign.
From
now, we imagine the surface referred to another parametric system, for which the linear element is
If,
(54)
ds*
= Edu + 2 Fdudv + G dv\
2
is given by (50) will be defined = const. And if the sign of $ be an equation such as by (u, v) such that is increasing in the direction of the normal of its pro
c/&gt;
&lt;
the curve whose geodesic curvature
jecting cylinder,
its
geodesic curvature will be given by
(55)
p.
where the differential parameters are formed with respect to (54). If two surfaces are applicable, and points on each with the same
curvilinear coordinates correspond, the geodesic curvature of the curve const, on each at corresponding points will be the same
&lt;=
in consequence of (55).
Hence
:
Upon two
applicable surfaces the geodesic curvature of corresponding
curves, at corresponding points, is the same.
136
GEOMETRY OF A SURFACE ABOUT A POINT
the second
When
member
of equation (55)
is
developed by
(III, 46, 56),
we have
1 1
d
\
du
dv\
d
\
dv
du
ff(V*d
cu
H
dv
R
du
d
L
cu
I
I
1
dv
1
H
F W\
d^\
d
I
du
dv\
dv
du,
Hence we have the formula
of
Bonnet*:
(56)
i^
du
dv
when
In particular, the geodesic curvature of the parametric curves, the latter do not form an orthogonal system, is given by
dv
ft.
(57)
du
The geodesic curvature
the differential equation
of a curve of the family, defined
by
has the value
1_
1
f
Mdu + Ndv*sQ,
d
I
2
"
pg
\ FN GM H \ du \^EN* 2 FMN+ GM /
d/
FMEN
ZFMN+ GM
*Memoire
Cahier 32
sur la theorie generale des surfaces, Journal de VEcole Poll/technique,
(1848), p. 1.
GEODESIC TOKSION
In illustration of the preceding results,
137
theorem
:
we
establish the
When
system
is
the curves of
an orthogonal system have constant geodesic
curvature, the
isothermal.
dsz
When the surface is referred to these lines, and = Edu2 + Gdv 2 the condition that the geodesic
,
the linear element is written curvature of these curves be
constant
is,
by
(50)
and
(51),
1
l/i,
dVE = fa
VEG
KI,
where Ui and V\ are functions of u and v respectively. If these equations are differentiated with respect to D and u respectively, we get
dudv
Subtracting,
dv
cu
we
is
obtain
""S
cucv
75^
dv
du
CUCV
log
G
=
0.
Hence
system
E/G
is
isothermic.
equal to the ratio of a function of u and a function of u, and the In terms of isothermic parameters, equations (i) are of
the form
10X =
\2 5U
r
r,
J.3X
\2 dV
~
and the
(II)
linear element
is
tf
It is evident that the
such a system. on a sphere.
The same
meridians and parallels on a surface of revolution form is true likewise of an orthogonal system of small circles
have just seen that when a curve is or a differential equation, its geodesic equation curvature can be found directly. The same is true of the normal
59. Geodesic torsion.
finite
We
denned by a
curvature of the surface in the direction of the curve by (18). Then from (16) and (47) follow the expressions for p and tw. In
order to define the curve
for the torsion.
it
remains for us to obtain an expression
it
From
(59)
the definition of
sin
o&gt;
follows that
o)
= X\ f Y/JL + Zv,
If this
where
X,
/Lt,
v are the directioncosines of the binormal.
equation is differentiated with respect to the arc of the curve, and the Frenet formulas (I, 50) are used in the reduction, we get
(
60 )
v
^
_i
*
ds
138
GEOMETRY OF A SURFACE ABOUT A POINT
(I,
From
37, 41) Ave
_
~~~
/dy
d"z
have
dz d*y\
_ ^~
/dz ~ d*x
_ dx ~~~~
d"*z d"*z\
~d~s~dfj
j
ds
2
ds ds
,
2
and
Moreover, from
r
r
d 2x
7
.
VK
/
.
6?S^
(13),
we
obtain the identity
ds ds
ds ds
H\_
\ds/
ds ds
Consequently equation
(61)
(60) is equivalent to
eos
where l/T has the value
1
"
 ED } du2 +(GD ~ (FD
2
ED")
dudv
+ GD ( 2
FD")
dv*
~T
H(Edu + 2 Fdudv + G dv
from
zero, that
is,
)
When
cos
w
is
different
when
the curve
is
not
an asymptotic
line,
equation (61)
becomes
hfF
As
cients
the expression for
and dv/du, we
T
involves only the fundamental coeffi have the following theorem of Bonnet:
is the
T
The function
tangent at a
ds
point.
same for
all
curves which have the same
common
Among
will be
these curves there
later
(
shown
one geodesic, and only one, for it one passes 85) that one geodesic and only
is
direction at the point. through a given point and has a given or 180, and conse of this geodesic w is equal to At every point of T for a given point and direc T. Hence the value T
quently
is
tion
The
that of the radius of torsion of the geodesic with this direction. function T is therefore called the radius of geodesic torsion of
GEODESIC TORSION
the curve.
139
From
(63)
it is
seen that
T
is
the radius of torsion of
makes a constant angle with the any curve whose osculating plane
tangent plane.*
When
is
the numerator of the righthand member of equation (62) of lines of equated to zero, we have the differential equation
curvature.
Hence
:
that the geodesic torsion of necessary and sufficient condition zero at a point is that the curve be tangent to a line of a curve be curvature at the point.
A
The geodesic
torsion of the parametric lines
1
is
given by
_ FDED
~T~
EH
_ GD l_ ~
~TV
FD"
GH
these lines form an orthogonal system Tu and Tv differ only in sign. Consequently the geodesic torsion at the point of meeting of two curves cutting orthogonally is the same to within the sign.
When
Thus
far in the consideration of equation (61)
the case of asymptotic lines. In considering that they are parametric. The directioncosines of the tangent and binomial to a curve v = const, in this case are
we have excluded them now, we assume
JL^,
~
=L^,
7=
;
+1 or 1. Consequently the directioncosines of the normal have the values principal
where
e is
and similar expressions
for
m
and
n.
When
in the Frenet formulas
d\
~~
I
dfju
_m
T
dv
ds
_n
r
ds
T
ds
we
substitute the above values, and in the reduction
make use
of (11)
(65)
*
and
(13),
we get
Thus
far exception
must be made
of asymptotic lines, but later this restriction will
be removed.
140
GEOMETRY OF A SURFACE ABOUT A POINT
In like manner, the torsion of the asymptotic lines u = const, is found to be V K. But from (64) we find that the geodesic torsiqn
in the direction of the asymptotic lines
(63)
is is qp
V
JT.
Hence equation
true for the asymptotic lines as well as for all other curves
on the surface.
Incidentally
we have
established the following theorem of Enneper
real asymptotic line at
:
equal to the abso lute value of the total curvature of the surface at the point; the radii of torsion of the asymptotic lines through a point differ only in sign.
is
The square of the torsion of a
a point
The following theorem
of Joachimsthal
is
an immediate consequence of
(63)
:
When two surfaces meet under a constant angle, the line of intersection is a line of curvature of both or neither; and conversely, when the curve of intersection of two surfaces is. a line of curvature of both they meet under constant angle.
For,
of T,
if
we
denote the values of w for the two surfaces, and Z\, T2 the values by subtracting the two equations of the form (63), that TI = T2 have,
o&gt;
&&gt;!,
2
,
which proves the
have
ds
(wi
uz)
first
part of the theorem.
Conversely,
if
\/T\
= l/T2 =
0,
we
=
0,
and consequently the surfaces meet under constant angle.
EXAMPLES
1.
Show
is
revolution
that the radius of geodesic curvature of a parallel on a surface of the same at all points of the parallel, and determine its geometrical
significance.
2.
Find the geodesic curvature of the parametric b a X = (u + v), y = (uv),
,
/
lines
on the surface
uv
&gt;
z
=
3.
Given a family of loxodromic curves upon a surface of revolution which cut
all
the meridians under the same angle a show that the geodesic curvature of curves is the same at their points of intersection with a parallel.
;
these
4.
5.
Straight lines on a surface are the only asymptotic lines which are geodesies.
Show
that the geodesic torsion of a curve
1
is
given by
0,
T
where
= I
1/1
1\ sin
.
*\fi
pj
)
2
6 denotes the angle which the direction of the curve at a point the line of curvature v = const, through the point.
6.
7.
makes with
Every geodesic
line of curvature is a plane curve.
line is
Every plane geodesic
a line of curvature.
it is
8.
When
If the
a surface
of curvature
9.
is cut by a plane or a sphere under constant angle, on the surface, and conversely.
a line
curves of one family of an isothermal orthogonal system have constant the same property. geodesic curvature, the curves of the other family have
SPHERICAL REPRESENTATION
60. Spherical representation.
erties of a surface
141
In the discussion of certain prop
of advantage to make a representation of * S upon the unit sphere by drawing radii of the sphere parallel to the positive directions of the normals to S, and taking the extrem
it is
S
ities of
the radii as spherical images of the corresponding points on a point moves along a curve on $, its image m describes a curve on the sphere. If we limit our consideration to a portion of the surface in which no two normals are parallel, the portions
S.
As
M
in a onetoone correspondence. the sphere is called the spherical map upon representation of the surface, or the Gfaussian representation. It was first employed by Gauss in his treatment of the curvature of
of the surface
and sphere will be
This
of the surface
surfaces. f
The
coordinates of
m
are the directioncosines of the
if
normal to
the surface, namely X, F, Z, so that
we put
the square of the linear element of the spherical representation
is
In
48 we established the following equations
:
(68)
_f FD GD dx FD ED % H du~ H du dX FD"GD ex FD
1
.
dx
dv
2
2
ED"
dx
dv
]
v
H
z
du
H
2
By means
tions
(o,
of these relations
and similar ones
in
F and
Z, the
func
c^,
& may
be given the forms
=
(69)
~ [GIf H
_
2
F(DD"
+ D + ED
2
)
D"],
JL
H
2
(
or, in
terms of the total and mean curvatures
52),
(70)
*
The sphere
of unit radius
and center at the origin of coordinates.
t L.c., p. 9.
142
GEOMETRY OF A SUEFACE ABOUT A POINT
may
be
In consequence of these relations the linear element (67)
given the form
(71) doz
Itm (Ddu+
2D dudv+D"dv
2 )
K(Edi? + 2 Fdudv
and, by (18),
(72)
^=(
(70)
\ J*
From
(73)
we have
also
ff.
e is is positive or negative. 1, according as Equations (69) are linear in E, F, G. Solving for the we have
where
K
latter,
E=
(74)
from the definition that the normals
In seeking the differential equation of the lines of curvature to the surface along such a
curve form a developable surface, we found ( 51) that for a dis placement in the direction of a line of curvature we have
fa
du
7
du
+
&
dv
dv
7
+ r(l^
X du + 2X dv \ = 3
,
,
.

0,
\du
2,
dv
/
and similar equations
o
in
y and
QT/"
where r denotes the radius of
If these equations be multi7}
"jf
principal curvature for the direction.
T7"
Q *7
&gt;
plied respectively
by

?.,
dv
_ and
dv
du
du
 and added, and
du
likewise by 
&gt;
dv
the an(l added, th resulting equations
may
be written
D du +D dv  r(fdu+di&gt;) = 0, D du +D n dv r(3du + gdv} = 0.
Eliminating
(75)
r,
we have
2
as the equation of the lines of curvature

"dudv
D
D"&dv*
=
0.
SPHERICAL REPRESENTATION
principal radii in the form
(76)
c^
(&lt;o
143
the equation of the Again, the elimination of du and dv gives
2
) r*(&lt;oD"
+ 3D
2
&D
)
r
+
(DD"
D = 0,
)
so that
(77)
These results enable us to write equations
(74) thus
:
(78)
61. Relations
between a surface and
its
is
spherical representation.
Since the radius of normal curvature
tion except when the surface following theorem
:
R
a function of the direc
is
a sphere,
we
obtain from (72) the
necessary and sufficient condition that the spherical representation of a surface be conformal is that the surface be minimal or a sphere.
A
As
a consequence of this theorem
is
onal system on a minimal surface
that every orthog represented on the sphere by
we have
an orthogonal system. From (70) it is seen that if a surface is not minimal, the parametric systems on both the surface and the sphere can be orthogonal only when If is zero, that is, when the lines of curvature are parametric. Hence we have
:
The
lines of curvature of
;
by an orthogonal system
this is
a surface are represented on the sphere a characteristic property of lines
of curvature, unless the surface be minimal.
This theorem follows also as a direct consequence of the theorem
:
necessary and sufficient condition that the tangents to a curve upon a surface and to its image at corresponding points be parallel is that the curve be a line of curvature.
A
In order to prove this theorem we assume that the curve
parametric, v
is
= const. Then
du du
the condition of parallelism
is
&.?.du
to
du
;&.*
du du
144
GEOMETRY OF A SURFACE ABOUT A POINT
(68)
it
From
But
follows that in this case
is
(FD
ED
1
)
must be
zero.
the latter
(
the condition that the curves v
= const,
be lines of
Moreover, from (32) it follows that the positive to a line of curvature and its spherical representa halftangents
curvature
51).
tion have the
same or contrary sense according
as the correspond
ing radius of normal curvature is negative or positive. In consequence of (7) the equation (40) of the asymptotic directions may be written
dxdX+ dydY+ dzdZ =
And
so
0.
we have
the theorem
:
The tangents
property
is
to
an asymptotic
line
and
to its
tion at corresponding points are
perpendicular
spherical representa to one another ; this
characteristic of asymptotic lines.
It is evident that the directioncosines of the
normal to the
sphere are equal to X, Y, Z, to within sign at most. be denoted by then I/, ^;
X&gt;&lt;&gt;
Let them
(79)
x&gt;
=
#\du
(
dv
?
du dv
When
in the parentheses,
expressions similar to (68) are substituted for the quantities the latter expression is reducible to KHX.
(73),
Hence, in consequence of
(80)
x&gt;
we have
= ex,
^
= er,
l
= *z,
is
where
e
=
1 according as the curvature of the surface
positive
or negative. From the above
is elliptic
it
follows that according as a point of a surface
or hyperbolic the positive sides of the tangent planes at corresponding points of the surface and the sphere are the same or
Suppose, for the moment, that the lines of curvature are parametric. From our convention about the positive direction of the normal to a surface, and the above results, it follows that both
different.
have the the tangents to the parametric curves through a point as the corresponding tangents to the sphere, or both is an elliptic point; but that have the opposite sense, when
M
same sense
M
one tangent has the same sense as the corresponding tangent to the sphere, and the other the opposite sense, when the point is
GAUSSIAN CURVATURE
145
hyperbolic. Hence, when a point describes a closed curve on a surface its image describes a closed curve on the sphere in the same or opposite sense according as the surface has positive or
say that the areas inclosed by these negative curvature. curves have the same or opposite signs in these respective cases. Suppose now that we consider a small parallelogram on the sur
face,
We
whose vertices are the points
(u
and
+
du, v
gram on
(u, v), (u f du, v), (u, v f dv), vertices of the corresponding parallelo the sphere have the same curvilinear coordinates, and
+ dv). The
the areas are Ifdudv and e/tdudv, where
e
1
according as the sur
face has positive or negative curvature in the neighborhood of the point (u, v). The limiting value of the ratio of the spherical and
the surface areas as the vertices of the latter approach the point a measure of the curvature of the surface similar to that (u, v) is
In consequence of (73) this limiting value is the Gaussian curvature K. Since any closed area may be looked upon as made up of such small parallelograms, we have the following
of a plane curve.
theorem of Gauss
The
:
limit of the ratio of the area of
a closed portion of a surface
to
the area of the spherical
is
image of it, as the former converges to a point, in value to the product of the principal radii at the point. equal
Since the normals to a developable surface along a generator are parallel, there can be no closed area for which there are not two nor
parallel. Hence spherical representation, as defined 60, applies only to nondevelopable surfaces, but so far as the preceding theorem goes, it is not necessary to make this exception
mals which are
in
;
for the total curvature of a developable surface is zero ( 64), and the area of the spherical image of any closed area on such a
surface
is zero.
The fact that the Gaussian curvature is zero at all points of a developable surface, whereas such a surface is surely curved, makes this measure not altogether satis factory, and so others have been suggested. Thus, Sophie Germain* advocated
the
mean
curvature, and Casorati
f
has put forward the expression 2
[
1
)
\Pi
fill
But according
a minimal surface is zero, and according minimal surface has the same curvature as a sphere. Hence the Gaussian curvature continues to be the one most frequently used, which may be due largely to an important property of it to be discussed later ( 64).
to the first, the curvature of
to the second, a
*
Crelle, Vol.
VII
(1831), p. 1.
f
Acta Mathematica, Vol.
XIV
(1890), p. 95.
146
GEOMETRY OF A SURFACE ABOUT A POINT
62. Helicoids. We apply the preceding results in a study of an important class of surfaces called the helicoids. A helicoid is generated by a curve, plane or twisted, which is rotated about a fixed line as axis, and at the same time translated in the direction
of the axis with a velocity which is in constant ratio with the section of the surface by a plane through velocity of rotation.
A
the axis
is
called a meridian.
All the meridians are equal plane
curves, and the surface can be generated by a meridian moving with the same velocities as the given curve. The particular motion described is called helicoidal motion, and so we may say that any
helicoid can be generated by a plane curve with helicoidal motion. In order to determine the equations of a helicoid in parametric
form,
we take
the axis of rotation for the 2axis, and let u denote
the distance of a point of the surface from the axis, and v the angle made by the plane through the point and the axis with the #zplane
in the positive direction of rotation. If the equation of the gen erating curve in any position of its plane is z (w), the equations
=
&lt;f&gt;
of the surface are
(81)
x
= u cos v,
y
= u sin v,
z
=
&lt;/&gt;
(u)
+ av,
;
where a denotes the constant
ratio of the velocities
it is
called
the parameter of the helicoidal motion. When, in particular, a is zero, these equations define any surface of revolution. Moreover,
when
&lt;
(u) is
a constant, the curves v
= const,
perpendicular to the axis,
and so the surface
is
are straight lines a right conoid.
It is called the right helicoid,
By
(82)
calculation
we
obtain from (81)
/2
c/)
,
F=a&lt;t&gt;
^= l +
,
G=u +a
2
2
,
where the accent indicates differentiation with respect to u. From method of generation it follows that the curves v = const, are meridians, and u = const, are helices on the helicoids, and circles
the
on surfaces of revolution. From (82) it is seen that these curves form an orthogonal system only on surfaces of revolution and on
the right helicoid. Moreover, from (57) it is found that the geo desic curvature of the meridians is zero only when a is zero or In the latter case the meridian is a straight line is a constant.
&lt;
perpendicular to the axis or oblique, according as
is
(/&gt;
zero or not.
HELICOIDS
Hence the meridians
of surfaces of revolution
147
and of the ruled
are
helicoids are geodesies.
The orthogonal
trajectories of the helices
(cf. Ill,
2
upon a helicoid
determined by the equation
afidu
31)
)
+ (u +
^
/
&gt;2
a
2
dv
= 0.
Hence,
if
we put
v1
=
J
It J
Q
2
du
+ v,
are the orthogonal trajectories, and their equations in finite form are found by a quadrature. In terms of the parameters u and v l the linear element is
the curves v x
= const,
2
(83)
t?s
u
of this result
As an immediate consequence
we have
that the
helices and their orthogonal trajectories on any helicoid form an isothermal system. From (83) and ( 46) we have the theorem of Bour:
Every
helices
helicoid
on the former correspond
We
(o4)
derive also the following expressions
,r
JL,
y,
z=
and
(85)
is
applicable
to
to
some surface of revolution, and the parallels on the latter.
:
a sin v
u&lt;f&gt;
cos v,
(a cos v
2
+ u$
2
!
sin
v),
u
V^
2
(l
+ f )+a
/&gt;,/&gt;
.!&gt;"=
From
(84) it follows that a meridian is a
its
face of revolution at all
curvature (Ex.
7, p. 140). of the lines of curvature of a helicoid,
normal section of a sur and consequently is a line of points, This is evident also from the equation
namely
]
(86)
a [1
+
2
&lt;/&gt;
+
u&lt;t&gt;
$
]
dii
2
 (1 + $ *) u^ + [(u + a +u +a ]dv =Q. a[u
2 2 )
M&lt;"
dudv
2

2
2
2
2
(t&gt;
Moreover, the meridians are lines of curvature of those helicoids,
for
which
&lt;/&gt;
satisfies the
condition
148
GEOMETRY OF A SURFACE ABOUT A POINT
integration this gives
By
=
&lt;f)
Vtf"
U2
C
log
When
and
D"
the surface
is
vanish.
Hence
the right helicoid the expressions for the meridians and helices are the asymp
D
totic lines.
Moreover, these lines form an orthogonal system, so that the surface is a minimal surface ( 55). Since the tangent planes to a surface along an asymptotic line are its oscu
lating planes,
if
the surface
is
a
ruled minimal surface, the gener ators are the principal normals of
all
the curved asymptotic lines. But a circular helix is the only
Bertrand curve whose principal
normals are the principal normals
of an infinity of curves
(
19).
Hence we have the theorem
Catalan
:
of
FIG. 16
The right helicoid is the only real minimal ruled surface.
In
fig.
16 are represented the asymptotic lines and lines of
is
curvature of a right helicoid. For any other helicoid the equation of the asymptotic lines
(87)
ufi du
2
 Zadudv + uty di? = 0.
As
the coefficients in (86) and (87) are functions of u alone,
:
we
have the theorem
helicoid is referred to its meridians and helices, the asymp totic lines and the lines of curvature can be found by quadratures.
When a
EXAMPLES
1.
Show
of revolution
2.
surface that the spherical representation of the lines of curvature of a is isothermal.
of a line of curvature
The osculating planes
and
of its spherical representa
tion at corresponding points are parallel.
the asymptotic directions at a point on a surface and as their spherical representation are equal or supplementary, according between at the point. the surface has positive or negative curvature
3.
The angles between
GENERAL EXAMPLES
4.
149
Show
that the helicoidal surface
x
is
=
u cos v,
y
=
u
sin v,
z
=
bv
minimal.
5.
The
total curvature of a helicoid
is
constant along a helix.
6.
The orthogonal
trajectories of the helices
upon a
helicoid are geodesies.
7. If the fundamental functions E, F, G of a surface are functions of a single parameter w, the surface is applicable to a surface of revolution.
8. Find the equations of the helicoid generated by a circle of constant radius whose plane passes through the axis and the lines of curvature on the surface also find the equations of the surface in terms of parameters referring to the meridians and their orthogonal trajectories.
;
GENERAL EXAMPLES
to a surface, and if 1. If a pencil of planes be drawn through a tangent to the sections of the surface by these lengths be laid off on the normals at planes equal to the curvature of the sections, the locus of the end points is a
M
MT
straight line normal to the plane determined by surface at M.
MT
and the normal
to the
2. If P is a point of a developable surface, P the point where the generator through P touches the edge of regression, t the length PoP, p and r the radii of curvature and torsion of the edge of regression, then the principal radii of the
surface are given by
i
= 0,
=
i
3.
For the surface
of revolution of a parabola about its directrix, the principal
radii are in constant ratio.
helices
The equations x = a cos it, y = asinw, z = uv define a family of circular which pass through the point A (a, 0, 0) of the cylinder each helix has an involute whose points are at the distance c from A (cf. I, 106). Find the surface which is the locus of these involutes show that the tangents to the helices are
4.
; ;
normal
5.
to this surface
;
find also the lines of curvature
upon the
latter.
The surfaces defined by
the equations
q*f(y),
(cf.
25)
l+p 2 + ?2 =
yaxis respectively.
6.
x
+
pz=&lt;t&gt;(p)
have a system of lines of curvature in planes parallel to the xzplane and to the
The equations
y
 ax =
0,
a:
2
+
y2
+
z2

2 px

a2
=
0,
where a and
(0,0, a). characteristics of a family of spheres, except when f(a) is a linear function also that the circles are lines of curvature on the envelope of these spheres.
7.
ft are parameters, define all the circles through the points (0, 0, a), Show that the circles determined by a relation ft=f(a) are the
;
If
the other nonrectilinear lines of curvature
one of the lines of curvature of a developable surface lies upon a sphere, lie on concentric spheres.
150
8.
GEOMETRY OF A SURFACE ABOUT A POINT
If
the center of normal curvature of the line is a point on a surface, the angle between the lines of curvature, and PI, P 2 the centers of normal bisecting curvature in two directions equally inclined to the first, then the four points
P
P
P, PI, PO, PZ form a harmonic range.
R m denote the radii of normal curvature of m sections R a Rs, which make equal angles 2 tr/m with one another, and m 2, then a surface
9. If EI,
, ,
&gt;
of
I /I
m
\Ri
10. If the
V1
R2
+ J_\ =
*
RJ
2
/i + T Vx
P
IV J
is
Dupin
indicatrix at a point
P of
a surface
an
ellipse,
and through
either one of the asymptotes of its focal hyperbola two planes be drawn perpen dicular to one another, their intersections with the tangent plane are conjugate
directions on the surface.
and whose osculating All curves tangent to an asymptotic line at a point 11 for a point of inflection. surface at 3f, have planes are not tangent to the
.
M
,
M
12.
The normal curvature
equal to the
13.
mean curvature
of an orthogonal trajectory of an asymptotic line of the surface at the point of intersection.
is
The surface
x
of revolution
whose equations are
u
sin w, z
=
u cos w,
y
=
=
a log (u
4
vV
2
a2 )
called the catenoid.
is
Show
a catenary about its axis generated by the rotation of that it is the only minimal surface of revolution.
;
it is
a constant angle 14. When the osculating plane of a line of curvature makes with the tangent plane to the surface, the line of curvature is plane. line of curvature is represented on the unit sphere by a circle. 15. A
plane
16.
The cylinder whose
tion p
=
a

s 2 /6,
it
where a and
lie
the intrinsic equa right section is the curve defined by b are positive constants, has the characteristic prop
erty that
I/Void.
17.
upon
curves of curvature
a
+
a26
b
,
^l ^
whose geodesic curvature
is
When
a surface
is
the curves v geodesic curvature of curvature of the curve the
referred to an orthogonal system of lines, and the radii of = const, and u  const. are p^, pgv respectively,
geodesic
which makes an angle
cos0
Pgu
sin
&gt;
Q
with the lines v
=
const,
is
given by
1
_ dd
~~
Pg
dS
Pgv
s referred to an orthogonal system of lines, and p vi for one system of isogonal denote the radius of geodesic curvature and the arc and pj, 8 the similar functions for the trajectories of the parametric lines, then whatever be the direction of the first trajectories of the former, orthogonal
18.
When
a surface
is
curves the quantity
19.
If p
4.
to
constant at a point.
first
and
p
denote the radii of
also p,
fi s
curvature of a line of curvature and
its
and spherical representation,
curves, then
and
ff
curvature of these p g the radii of geodesic
fa
dtr
77
where ds and
d&lt;r
=d
P~8~P?
are the linear elements of the curves.
GENERAL EXAMPLES
,
151
20. When a surface is referred to its lines of curvature, and #o denote the angles which a curve on the surface and its spherical representation make with the curves v = const., the radii of geodesic curvature of these curves, denoted by
pg
and
pg respectively, are in the relation
ds
ddo
Py
dcr = d&o 
21.
When
Pg
the curve
x=f(u)cosu,
is
y=f(u)sinu,
z
=and also geodesies
subjected to a helicoidal motion of parameter a about the zaxis, the various
positions of this curve are orthogonal trajectories of the helices, on the surface.
22. When a curve is subjected to a continuous rotation about an axis, and at the same time to a homothetic transformation with respect to a point of the axis, such that the tangent to the locus described by a point of the curve makes a con
stant angle with the axis, the locus of the resulting curves is called a spiral surface. Show that if the zaxis be taken for the axis of rotation and the origin for the center
of the transformation, the equations of the surface are of the
form
z
x
= f(u) e hv cos (u +
a constant.
spiral surface
line,
v) ,
y
= f(u) e hv sin (u +
v) ,
=
v (u) e* ,
where h
23.
is
can be generated in the following manner: Let C be a on C describes an a point on the latter if each point isogonal trajectory of the generators on the circular cone with vertex P and axis I in such a way that the perpendicular upon I, from the moving point revolves about I
curve,
I
A
any
and
P
;
M
M
,
with constant velocity, the locus of these curves
24.
is
a spiral surface
(cf
.
Ex.
5,
33).
that the orthogonal trajectories of the curves u = const., in Ex. 22, can be found by quadratures, and that the linear element can be put in the form
Show
where
A
is
a function of
a
alone.
25. Show that the lines of curvature, minimal lines, and asymptotic lines upon a spiral surface can be found by quadrature.
CHAPTER V
FUNDAMENTAL EQUATIONS. THE MOVING TRIHEDRAL
63. Christoffel symbols.
and
sufficient equations of condition to be satisfied
In this chapter we derive the necessary by six func
tions, E, F,
G
;
D, D\
D",
in order that they
may be
of
the fundamental
quantities for a surface.
For the sake
of brevity
we make use
two
sets of symbols,
suggested by Christoffel,* which represent certain functions of the coefficients of a quadratic differential form and their derivatives of
the
first order.
If the differential
form
is
a n du 2 f 2 a^du^du^ +
the
first
c
set of symbols
is
defined by
R&l
[l J
!/^ + ^_&lt;HA
2\du k
i,
du
(
duj
where each of the subscripts
k,
I
has one of the values 1 and 2.f
From
this definition it follows that
When these symbols are used in connection ,with the first fun 2 2 2 damental quadratic form of a surface ds = E du + 2F dudv + G dv they are found to have the following significance
:
,
[iriia*
L1J
2Su
2dv
[""L^ia* 2 Su L2J
(1)
2J O & cu
,*
2
di&lt;
I
~l
I
L
* Crelle, Vol.
t
.
J
I
cv
Q/*
O L^J
I
I
^ 2~dv
LXX,
pp. 241245.
This equation defines these symbols for a quadratic form of any number of vari n. u n In this case i, k, I take the values 1, ables wi, 152
,
,
CHRISTOFFEL SYMBOLS
The second
set of
153
symbols
is
defined by the equation
where A vl denotes the algebraic complement of a vl in the discrimi a 22 divided by the discriminant itself. With reference nant a n a 22 to the first fundamental quadratic form these symbols mean
and
4iJv
du
f)
^ =:^
du
A
=!r*
dv
TT 2
fill
I
du
,"
dv
TiO
du
n
I
ri2\
^g^
du
llJ
f"
2^
G
^v
3t*
I 12 \_
2
du
dv
l2/~
f221
dv
G
dv
du
dv
From
these equations
we
derive the following identities
:
With
the aid of these identities
we
derive from (III, 15, 16) the
expressions
&gt;
(MVKff}&gt;
From
the above definition of the symbols
:
SmK?))f^l
&lt;!
&gt;
WQ
obtain the
following important relation
64. The equations of Gauss and of Codazzi. equations (IV, 10) and the equation
The
first
two of
154
FUNDAMENTAL EQUATIONS
set of equations linear in
2
,
form a consistent
determinant
is
du
2
,
du 2
du
2
and the
equal to H.
Solving for
,
we get
similar equations hold for
y and z. Proceeding the other equations (IV, 10) and
(6)
in like
manner with
we
get the following equations of Gauss
:
a
2
*
_ri2\az
fiaiat
awa
li/a*
la/a*
^
For convenience of reference we
recall
from
48 the equations
dx
dv
dX_FD =
~du
(8)
GDdx
H
2
a^
4
FDED
n
FD
2
dX __ ~
3v
FD"GD dx
ED"
dx
dv
(7) are
H
2
du
H
2
The
conditions of integrability of the Gauss equations
du\dudv
dv \dudv
du\dv
2
By means
(9)
of (7)
and
&lt;*
(8)
these equations are reducible to the forms
x
1
a
v
o
2
^ 4. A
du
?E
2
_i_
x
;
1
dv
v
,
D"
D, D c 2 are determinate functions of E, F, G where a v a 2 similar to (9) hold for y and and their derivatives. Since equations z, we must have
,
&gt;,
(10)
!=0,
a 2 =0,
&j=0,
62
=0,
^=0,
&lt;?
2
=0.
EQUATIONS OF GAUSS AND OF CODAZZI
When
,
155
the expressions for a r a 2 b^ and 5 2 are calculated, it is found that the first four equations are equivalent to the following :
12 12\
d
jMlll
d
12jfl2j
,
fill
J221
/in
ri2i
rni/121
ri2i
2
,
rnif22
fi2i fill
221
HiAaJtaJ
d
M21
,
/221 /121
,
/221
Ml
f!2\
a
/22\
,
fl2\fl2\
f22\/ll
When
the expressions for the Christoffel symbols are substituted
in these equations the latter reduce to the single equation
Z?D"jP
a
H
=
1
f
gr
F
dF
du
2
~2N \du \_EH
j^r
dv
dE__ 1 3G1
dv
H du\
_
2
i
IH
H
dv
EH
du_
This equation was discovered by Gauss, and is called the Gauss equation of condition upon the fundamental functions. The left
hand member
of the equation is the expression for the total curva ture of the surface. Hence we have the celebrated theorem of Gauss *
:
The expression for the total curvature of a surface is a function of the fundamental coefficients of the first order and of their deriva tives of the first and second orders.
When
the last
the expressions for
c^
and
c2 are calculated,
we
find that
two
of equations (10) are
v
(13)
"
du
dv
*L.c., p. 20.
156
FUNDAMENTAL EQUATIONS
These are the Codazzi equations, so called because they are equiva lent to the equations found by Codazzi * ; however, it should be mentioned that Mainardi was brought to similar results some
what
earlier. f
It is
sometimes convenient to have these equations
written in the form
D~
(13
)
d
D
,
f22\D
"
f!2\D
fin
~~
which reduce readily to
(13)
by means of
(3).
With
to (13).
the aid of equations (7)
we
find that the conditions of
integrability of equations (8)
and similar ones in
Y
and
Z
reduce
From
surfaces
the preceding theorem and the definition of applicable ( 43) follows the theorem
:
Two
the same total curvature at corre applicable surfaces have
sponding points.
As
a consequence
we have
:
Every surface applicable
twisted curve.
to
a plane
is the
tangent surface of a
For,
when
a surface
2
is
applicable to a plane
its
linear element
is
reducible to ds
is
=
du
2
+ dv\
and consequently
its
total curvature
zero at every point by (12).
2
From
(IV, 73) it follows that
Hence X,
Y,
Z
the surface
is
are functions of a single parameter, and therefore the tangent surface of a twisted curve (cf. 27).
Incidentally
we have proved
the theorem
:
When
and
K
is zero at all
the latter points of a surface
is
developable,
conversely.
una
Vol. II superficie e dello spazio, Annali, Ser. 3,
p. 395.
* Sulle coordinate curvilinee d
(18(W)
t
,
p. 269.
Giornale dell Istituto Lombardo, Vol. IX,
FUNDAMENTAL THEOREM
65.
157
Fundamental theorem. When the lines of curvature are and Codazzi equations (12), (13) reduce to parametric, the Gauss
DP"
_
(14)
G
2_ (
dv
D
"\
_^ E
~
du
The
v
directioncosines of the tangents to the parametric curves,
= const,
and u
= const.,
have the respective values
(15)
2
By means
of equations (7)
and
(8)
we
find
D"
(16)
.
du
V
ax
(
du
and similar equations obtained by replacing X^ X^ Z. From (15) we have respectively, and by Z^ Z^,
X by Yv T Y
2
,
= C^EX
(17)
= CVflY = CVEZ
We
proceed to the proof of the converse theorem
D",
:
Given four functions, E, G, D, exists a surface for which E, 0,
quantities of the first
G
;
satisfying equations (14); there 0, D" are the fundamental
Z&gt;,
and second order
respectively.
158
In the
FUNDAMENTAL EQUATIONS
first place we remark that all the conditions of integraof the equations (16) are satisfied in consequence of bility (14). Hence these equations admit sets of particular solutions whose
values for the initial values of u and v are arbitrary. From the form of equations (16) it follows (cf. 13) that, if two such sets
of particular solutions be denoted
by X^
2
X X and Y^ F
z,
2,
F,
then
XI + XI + X* =
2
const.,
I
0, 0, 1.
Y?+ F + F = const., X F + X Y + XY= const.
1 x
2
2
From the theory of differential equations we know that there exist three particular sets of solutions X^ X\ Fp F2 F; Z^ Z^ Z, z which for the initial values of u and v have the values 1, 0, 0; 0, 1, 0;
X
,
,
In this case equations (18) become
X + X + ^ = 1,
2
1
2 2
2
2
(19)
+F + F = X Y +X Y + XY=Q,
F
2
2
2
l,
i
l
2
2
which are true
for all values of
u and
v.
In like
manner we have
(19
)
follows that the expressions in the righthand mem bers of (17) are exact differentials, and that the surface denned by these equations has, for its linear element and its second quadratic
From
(16)
it
form, the expressions
(20)
Edu?+G dv\
we had
D dy? +
2
D"dv
respectively.
Suppose, now, that
a second system of three sets of
solutions of equations (16) satisfying the conditions (19), (19 ). s, F s, and Z s equal By a motion in space we could make these to the corresponding ones of the first system for the initial values
X
of
u and v. But then, because of the relations similar would be equal for all values of u and v, as shown in
motion in space, a surface
(20).
is
to (18), they
to within a
ratic
13. Hence, determined by two quad
forms
As
in
13,
it
can be shown that the solution of
equations (16) reduces to the integration of an equation of Riccati.
FUNDAMENTAL THEOEEM
159
Later * we shall find that the directioncosines of any two per
and of the normal a pendicular lines in the tangent plane to surface, to the surface, satisfy a system of equations similar in form to (16). Moreover, these equations possess the property that sets of solu
tions satisfy the conditions (18) when the parametric lines are any whatever. Hence the choice of lines of curvature as parametric
lines simplifies the preceding equations,
but the result
is
a general
one.
Consequently we have the following fundamental theorem:
the coefficients of two quadratic forms,
When
Edu* +
to
2 Fdudv
+ G dv\
Ddu +2 D dudv +
2
D"dv\
satisfy the equations of
Gauss and Codazzi,
there exists a surface,
unique
within
its
respectively the first
forms are and second fundamental quadratic forms ; and
a Riccati
which these position in space, for
the determination of the surface requires the integration of
equation and quadratures.
are the funda From (III, 3), (5) and (6), it follows that if E, F, G; D, D mental functions for a surface of coordinates (x, y, z), the surface symmetric with the coordinates ( x, y, z), has respect to the origin, that is, the surface with
,
D"
 D". Moreover, in consequence the fundamental functions E, F, G;  D, two surfaces whose fundamental quantities bear such a rela of the above theorem,
D
,
tion can be
Two
moved in space so that they will be symmetric with respect to a point. surfaces of this kind will be treated as the same surface.
EXAMPLES
1.
When
is
surface
2.
said to be isothermic.
the lines of curvature of a surface form an isothermal system, the Show that surfaces of revolution are isothermic.
Show
that the hyperbolic paraboloid
x
is
=a + (t*
is
t&gt;),
y
b = (*),
*
= uv
in terms of
isothermic.
3.
When
a surface
isothermic,
and the linear element, expressed
is
parameters referring to the lines of curvature, of Codazzi and Gauss are reducible to
Pl
i
ds 2
=
\2 (du 2
+
dv 2 ), the equations
dp 2
a
_ PZ
i
api
Find the form of equations 39). symmetric coordinates (cf
4.
.
(11), (13)
when
the surface
is
defined in terms of
* Cf
.
69.
Consult also Scheffers, Vol.
II,
pp. 310 et seq.
;
Bianchi, Vol.
I,
pp. 122124.
160
FUNDAMENTAL EQUATIONS
K
is equal to zero for the tangent surface of a twisted 5. Show that curve, taking the linear element of the latter in the form (105), 20.
6.
Show
its
that the total curvature of the surface of revolution of the tractrix
about
7.
axis is negative
and constant.
Establish the following formulas, in which the differential parameters are formed with respect to the form Edu? + 2Fdudv + Gdv 2
:
)=~where the quantities have the
8.
same
significance as in
JT,
65.
Deduce the
identity
A2x =
(
1
)
and show therefrom that the curves in which a minimal surface is cut by a family of parallel planes and the orthogonal trajectories of these curves form an isothermal system.
66.
Fundamental equations
if
in another form.
We
have seen in
61 that
X, F,
Z denote
the directioncosines of the normal to
a surface, the directioncosines of the normal to the spherical rep 1 according resentation of the surface are eX, eF, eZ, where e is
as the curvature of the surface
is
positive or negative.
If,
then,
the second fundamental quantities for the sphere be denoted by
A^
(21)
,
3",
we have
=&lt;
^ = e^,
,&"
= e^
so that for the sphere equations (7)
become
rnvax_
2X
(22)
J
+
12J dv 2 f 12V
l2J
F
^x;
where the Christoffel symbols
T
\
?\
are formed with respect to
the linear element of the spherical representation, namely
conditions of integrability of equations (22) are reducible by means of the latter to
The
= 0,
EQUATIONS OF CODAZZI
where
a v a2
,
161
A.,
A 2J B^ B
b lt b 2
,
are the functions obtained from the quantities 2 DD"D 64 by replacing of E, F, G respectively
2
&gt;
by
&, ^respectively. Since the above equations must be sat isfied by Y and Z, the quantities A^ A 2 B v Bz must be zero. This
1,
,
gives the single equation of condition
(24} J
J_rA/^^__L^V^ fa  2v ^ du)\ = 1 ^1 ft 2 ft to &ft dv ft du
[du
\ft
)
W
Moreover, the Codazzi equations (13
of (21),
)
become, in consequence
(26).
3w
o
\
W
//"/
o
/
v \///
V
//
/
1 \.\jft
I
//"
~
*
~*
*
f
~*
*
""
which vanish
If
identically.
dx
equations (IV, 13) be solved for
f
and
dx
cv
we
du
get
dx
_
"
.
(26)
i_
dv
ft*
du
By means
of equations (22) the condition of integrability of these
g
equations, namely
/^\
^
and similar conditions in y and
2,
reduce to
(27)

OU
^Hu
v^
dv
Hence two quadratic forms
(odu
2
f
2
& dudv +
dv
2
,
D du
2
+2Z&gt;
dudv
+
2
D"dv
,
whose
coefficients satisfy the conditions (24), (27), may be taken as the linear element of the spherical representation of a surface
and as the Si3cond quadratic form of the
latter.
When X,
F,
Z are
162
FUNDAMENTAL EQUATIONS
of the surface can be found by however, the determination of the former requires
known, the cartesian coordinates
;
quadratures (26) the solution of a Riccati equation.
If the
equations
t
D = _^fc*x
by
(7)
be differentiated with respect to u and to the form * and means of
(22)
:
v,
the resulting equations
may
be reduced
55
cu
(ii
(
11
1
2
12
l
=
a
I2
[
2) f
c
i
Jl
2
cD
^=
cu
cv
*
JX +
D+
12
2
22
)
D
cu
f
2
D
cv
D"
D".
(V
surface may be 67. Tangential coordinates. Mean evolute. not only as the locus of a point whose position looked upon but also as the envelope of its two
A
depends upon
parameters,
tangent planes. the surface is developable or not. We parameters according as case in 27, and now take up the latter. considered the former distance from the origin to the tan denotes the algebraic If
This family of planes depend* upon one or two
W
S at the point M(x, gent plane to a surface
(29)
y, z),
then
W=xX+yY+zZ.
with respect to u and
v,
If this equation is differentiated
the
to
in consequence of (IV, resulting equations are reducible,
3),
X
dW
I, p.
*Cf. Bianchi, Vol.
157.
TANGENTIAL COORDINATES
The
163
three equations (29), (30) are linear in #, y, z, and in con Hence sequence of (IV, 79, 80) their determinant is equal to e/
we have
and similar expressions for y and
identities
z.
From
(IV, 11)
we deduce
the
Y
(
dz
du
a
Z
dY
du
y.
e = TH/
^dX
ex
1
,
r
~dX\
&lt;o
01 dl )
r
rf \
,
du
C^\
dv /
zr&lt;
)
rC
rrV^
v
I
~Y
j^GA
By means
of
these
equations the
above expression for x
is
reducible to
cv
Hence we have
(32)
x
= WX+k[(W,X),
y=
WY+k((W,Y),
z
= WZ+ &((W,Z],
the differential parameters being formed with respect to (23). of u and Conversely, if we have four functions X, F, Z,
W
i&gt;,
such that the
(33)
first
three satisfy the identity
x +r +^ = l,
2
2
2
equations (32) define the surface for which X, F, Z are the directionis the distance of the latter from cosines of the tangent plane, and the origin. For, from (33), we have
W
= 0,
dv in consequence of
which and formulas
&gt;TA
(22)
we
find
from
(32) that
dx
Moreover, equation (29) also follows from (32). Hence a surface is completely defined by the functions X, F, Z, W, which are
called the tangential coordinates of the surface.*
* Cf Weingarten, Festschrift der Technischen Hochschule zu Berlin (1884) Darboux, Vol. I, pp. 234248. I, pp. 172174
. ;
;
Bianchi,
Vol.
164
FUNDAMENTAL EQUATIONS
equations (30) are differentiated,
When
we
obtain
ffw
_
dv*
By means
of (22), (29),
and
(30) these equations are reducible to
du
(34)
2
D =
tfW
\_dudv
are substituted in the these expressions for D, for p^+ p 2 the latter becomes expression (IV, 77)
When
A
&"
,
By means
(35)
of (25) this equation can be written in the
/&gt;
form
1
4^ 2 =
(A;TF42^),
is
where the
differential
parameter
formed with respect
to the
linear element (23) of the sphere. Moreover, if A^ 2 denotes the following expression,
_ r22y^__
1
1 J an
2J
12
follows from (34) that
12
it
(3T)
MEAN EVOLUTE
In passing
165
a differential parameter
we
shall prove that
it is
A 22
is
by showing that
(38)
expressible in the
form
Without
(39)
loss of generality
we
take
2
Edu*+Gdv
Then
/I dG
1
as the quadratic form, with respect to
which these
differential
parameters are formed.
1
1
dE\
1
dE
u
=
F
dv
\du
By
substitution
we
find
_
eters, their
Since the terms in the righthand member are differential param values are independent of the choice of parameters
v,
u and
is
in terms of
which
(39)
is
expressed.
Hence equation
(38)
an identity.
The
coordinates #
face halfway
y Z Q of the point on the normal to a sur between the centers of principal curvature have
,
,
the expressions
The
is
surface
enveloped by the plane through this point, which
is
parallel to the tangent plane to the given surface, mean evolute of the latter.
If
called the
W
denotes the distance from the origin to this plane,
we have
(40)
W,= ZXf9 =W+^(p +Pt ).
1
By means
(41)
of (35) this^may be written
TFO =JA;TF.
166
FUNDAMENTAL EQUATIONS
EXAMPLES
Derive the equations of the lines of curvature and the expressions for the principal radii in terms of W, when the parametric lines on the sphere are
1.
(i)
meridians and parallels
;
(ii)
the imaginary generators.
lie
Show that in the latter case the curves corresponding to the generators metrically with respect to the lines of curvature.
sym
2. Let Wi and 2 denote the distances from the origin to the planes through the normal to a surface and the tangents to the lines of curvature v = const. , u = const, respectively, so that we have
W
Show
Wi = xX l + yYi +
that
zZi,
W
2
= xX2 + yYz + zZ 2
Pi
.
the differential parameters being formed with respect to
3.
Edu* +
2
Fdudv +
If 2 q
=
x2
+
yz
+
z2,
then
we have
4.
Show
that
when
the lines of curvature are parametric
=
Pi
cu
cu
~ =
P2
v
dv
is
5. The determination of surfaces whose mean evolute is a point problem as finding isothermal systems of lines on the sphere.
the
same
dition
trihedral. The fundamental, equations of con be given another form, in which they are frequently may used by French writers. In deriving them we refer the surface to
68.
The moving
a moving set of rectangular axes called the trihedral T. Its ver tex is a point of the surface, the a^plane is tangent to the surface at M, and the positive 2axis coincides with the positive
M
direction of the normal to the surface at
x
M.
and
?/axes is
zaxis, U being a function of u and v. given In Chapter I we considered another moving trihedral, consisting of the tangent, principal normal, and binormal of a twisted curve.
the curve v
= const,
determined by the angle U makes with the through
M
position of the which the tangent to
The
Let us associate such a trihedral with the curve
v
const, through
THE MOVING TRIHEDRAL
16T
M and
we
call
call it the trihedral
t
u.
We have found
,
tions of the directioncosines a
6
,
c
16) that the varia ( of a line L, fixed in space,
with reference to
(7M ,
M,
as its vertex
moves along the curve which
are given
b
by
(42 )
,
&lt;^=
ds u
pu
f W +
dsu
\ Pu
;
where p u r u denote the radii of first and second curvature of Cu1 and dsu its linear element evidently the latter may be replaced
,
by
V^ du.
1
The
directioncosines of
r
L
with respect to the trihedral
&gt;
T
have
the values
(43)
a
_a
a
&gt;
Ib
_ ^ s n ^ _ c cos c cos sin U + (b sin w u
cos
jj
j
r
^
o&gt;
s j n ^r
?7,
M ) cos
=6
[&lt;7
f
cos
o&gt;
w f
c sin
o)
tt ,
where w u
makes
the angle which the positive direction of the zaxis with the positive direction of the principal normal to Cu at Jf,
is
the angle being measured toward the positive direction of the binormal of Cu From equations (42) and (43) we obtain the following
.
:
(44)
da 
= br
,
db
cq,
du
j9, q,
du
= cp
do
ar,
=
:
,
cu
aq
op,
where
r
have the following significance
p=
(45)
cosU
rr/ sm7(
c?&&gt;
1
.
)f sin
U coso)
coso).
=
1\
2
I
rr cosU
ds,.
V
If,
in like
u
= const,
manner, we consider the trihedral tv of the curve through M, denoted by Cv we obtain the equations
,
da
,
db
do
,
where
^^
Pui
* q v r l can be obtained from (45) by replacing Vjg; Z7, M , S ^ denotes the angle which the Tu ^7 ^^^ ^ 8 Pv TV
pv

v&gt;
i;
A
tangent to the curve
(46)
Cv
at
M makes with the
VU=G.
a&gt;axis,
we have
168
If the
line,
FUNDAMENTAL EQUATIONS
vertex
is,
that
along a curve other than a parametric along a curve determined by a value of dv/du, the
,
M moves
c are
variations of a,
evidently given by
^
da du
da dv
dv ds
^_
dv ds
do
du
do dv dv ds
du ds
in
du ds
du ds
which the
69.
differential quotients
have the above values.
Fundamental equations
with the trihedral
ciate
T
Suppose that we asso a second trihedral TQ whose vertex is
of condition.
it
fixed in space, about
which
revolves in such a
manner that
its
edges are always parallel to the corresponding edges of T, as the vertex of the latter moves over the surface in a given manner.
The position of T is completely determined by the nine directioncosines of its edges with three mutually perpendicular lines L v L 2
,
L
s
Call these directioncosines a v b { c l through These functions must satisfy the equations 3
0.
,
&lt;?
;
a2 5 2
,
,
c
3
,
.
da 
= 6r,
^w
(47)
dv~
If
= br TI
the
we equate
make use
cucv
two
tion
of these equations,
and
in the reduction of the resulting
equa
of (47),
we
find
Since this equation must be true
b
,
&lt;?
when
b
and
c
have the values
;
5
,
**
;
63 ,
&lt;?
8,
the expressions in parenthesis T
^&gt;O
must be equal
/^9
to zero.
Proceeding in the same manner with obtain the following fundamental equations *
:

and
d*c
we
dudv
dp
dpi
dq
(48)
dq l
dr
dr l
Ser.
* These equations were first obtained by Combescure, Annales de VEcole Normale, 1, Vol. IV (1867), p. 108; cf. also Darboux, Vol. I, p. 48.
ROTATIONS

169
,
These necessary conditions upon the six functions p, r 1? in cs may determine the order that the nine functions a x position of the trihedral T are also sufficient conditions. The proof of this
, , ,
is
similar to that given in
65.
* Equations (47) have been obtained by Darboux from a study of the motion of the trihedral TQ He has called jt?, q, r l the
.
rotations.
We
t
u
.
Let
return to the consideration of the moving trihedrals T and ?/, z ) denote the coordinates of a point P (x, y, z) and (#
,
with respect to
u respectively. the following relations hold
:
T
and
t
Between these coordinates
/
=x =#
f
cos
U
&lt;W
(y sin
o&gt;
M
z cos 2 cos
W M ) sin
&&gt;
7,
sin CT
1
(x
z If in a
y cos
+ (# M+ 2
sin
Wu
o&gt;
M)
cos
/,
sin
M.
the trihedral
these
displacement of P absolute increments with respect to t be indicated by S, and increments relative to u at
M
moving axes by
c?,
we
have, from
16,
^L^S^+i, d ds
u u
pu
dsu
(45)
= ^_ +  + ,
dsu
pu TM
^. =
C?S M
:
dz
Ly.
ru
ds u
From
(49), (50),
and
$x
we
obtain the following!
du
= dx 4 VrE cos Ury + qz,
du
^
^W
2
=
^M
+ VfismUpz + ra,
ox
=
dz
aw
a%
+ PV.
Equations similar to these follow also from the consideration t Hence, when the trihedral T moves over the v surface with its vertex describing a curve determined by a
of the trihedral
.
M
value of dv/du, the increments of the coordinates of a point P(x, y, z), in the directions of the axes of the trihedral, in the
I, chaps, i and v. In deriving these equations we have made use of the fact that equations (49) define a transformation of coordinates, and consequently hold when the coordinates are replaced by the projections of an absolute displacement of P.
*
t
L.c., Vol.
170
FUNDAMENTAL EQUATIONS
may
also be
absolute displacement of P, which * to these axes, have the values
moving
relative
where we have put
The
coordinates of
M are
(0, 0, 0),
so that the increments of
its
displacements are
(53)
Sx
=
1;
du
+
^dv,
y
= vidu +
,
ri l
dv,
Sz
= 0.
with respect to the y v 2J denote the coordinates of L 3 previously defined, it the lines L v fixed axes formed by 2
If fa,
M
follows that
and similar expressions
ag
,
for
y and z^ where a
l
t,
6 1?
^;
2,
62 ,
2
6 8 , c 3 are
to the
moving
the directioncosines of the fixed axes with reference axes. Since the latter satisfy equations (47), the
conditions that the
a
2
two values
*
dz
z
of
cu
^ obtained from (54) be equal,
and similarly for J
and 1 are
dudv
(55)
have ten functions f f p ?/, 77^ p, p^ q^ r, r satis , c s can be and (48), the functions a 1? fying these conditions of a Riccati equation, and x^ y# z l by quad found by the solution as well as ratures. Hence equations (48) and (55) are sufficient to the Gauss and are equivalent necessary, and consequently Codazzi equations.
When we
,
&lt;?,
x,
* Cf.
Darboux, Vol.
II, p. 348.
LINES OF CURVATUKE
70. Linear element.
171
(53)
Lines of curvature.
is
From
we
see that
the linear element of the surface
(56)
ds
2
= (%du + ^ dvf +(ndu + r
)i
dv}\
Hence a necessary and
lines be orthogonal is
sufficient condition that the parametric
(57)
ff!+^i==0.

c), it being For a sphere of radius c the coordinates of the center are (0, 0, that the positive normal is directed outwards. As this is a fixed point, it assumed follows from equations (51) that whatever be the value of dv/du we must have
and consequently
/KQ\
+ hdv  (qdu + qidv)c = ydu + dv + (pdu + pidv)c =
du
171
0,
0,
zr
i)
q
P
=
1
=
^i
=: C.
qi
Pi
in space, c) is fixed Conversely, when these equations are satisfied, the point (0, 0, therefore the surface is a sphere. Moreover, suppose that we have a propor and tion such as (58), where the factor of proportionality is not necessarily constant. For the moment call it t. When the values from (58) are substituted in (55) and
reduction
is
made
in accordance
dt
r?
with
dt
(48)
we
get
dt
l
7?1
^"
^
=
3t
*to~*
to
31, is seen to
 f^ is zero, which, from (56) and t is constant unless be possible only in case the surface is isotropic developable.
Hence
^
By
definition
(51)
a line of curvature
is
a curve along
which
the
the normals to the surface form
a developable surface.
When
vertex
is
move
in
a point (0, 0, p) must displaced along one of these lines, are zero. Hence we must have such a way that Bx and
%
^dv + (qdu + q dv r]du + (pdu+p
f du
h
i] 1
l
dv)p
l
= 0, dv)p = 0.
the equation of Eliminating p and dv/du respectively, we obtain the lines of curvature,
(59) (f
du
+ ^dv) (p du + p
z )
v
dv)
+
(17
du
+ rj^v) (q du +
)
q^v)
= 0,
0.
and the equation of the principal
(60)
radii,
p
 qPl + p (qrj,  q,rj + p^  p (pq,
+ (fa  ^) =
From (59) it follows that a necessary and sufficient condition that the parametric lines be the lines of curvature is
(61)
fe&gt;
+ i# = 0,
f^i+ih^O.
172
FUNDAMENTAL EQUATIONS
may
replace these equations by
\rj,
We
P=
?
= xf,
^^V?!,
?i
= \fn
When
these
thus introducing two auxiliary functions X and \. values are substituted in the third of (55), we have
X and \ are equal, the above equations are of the form (58), which were seen to be characteristic of the sphere and the isotropic developable. Hence the second factor is zero, so that equa
If
tions (61) (62) or
(63)
may be
replaced by
ffi+
^=0,
0,
m+??i=&lt;&gt;
^=17 =
(52)
it
P=
q1
=Q
From
follows that in the latter case the x and ?/axes are
tangent to the curves v
this case later.
= const,
we
and u = const.
We
shall consider
From
(60)
and
(52)
find that the expression for the total
curvature of the surface
is
where
denotes the angle between the parametric curves. the third of equations (48) may be written
co
Hence
/g4\
V76rsin
PiP*
co
H
PiP
dr
dr.
71. Conjugate
directions
representation.
We
and asymptotic directions. Spherical have found ( 54) that the direction in the
tangent plane conjugate to a given direction is the characteristic of this plane as it envelopes the surface in the given direction. Hence, from the point of view of the moving trihedral, the direc
is
tion conjugate to a displacement, determined by a value of dv/du, the line in the #?/plane which passes through the origin, and
which
does
not
experience
2;axis.
an
absolute
displacement in the
it
direction of the
From
the third of equations (51)
is
is
seen that the equation of this line
(65)
(p du
H
p^v) y
(q
du
+ q^dv) x =
0.
CONJUGATE DIRECTIONS
If the
173
increments of u and
v,
corresponding to a displacement in
the direction of this line, be indicated by d^ and d^v, the quan tities x and y are proportional to (f d^u f ^v) and (r; d^ 4 rj^v). When x and ?/ in (65) are replaced by these values, the resulting
+
equation
(66)
may
be reduced to
]l
 gf) dudjU + (pr  qgj dudy + (p&  qg) d^udv (prj
In consequence of (55) the coefficients of dud^v and d^udv are equal, so that the equation is symmetrical with respect to the
two
tion
sets of differentials, thus establishing the fact that the rela
between a line and its conjugate is reciprocal. In order that the parametric lines be conjugate, equation (66) must be satisfied by du = and d^v = 0. Hence we must have
(67)
It
should be noticed that equations (61) are a consequence of the
of (62)
first
and
(67).
Hence we have
the result that the lines of
curvature form the only orthogonal conjugate system. From (66) it follows that the asymptotic directions are given by
(68)
(prj
 gf
)
du* +
(prj l
 q^ +p^  q) dudv + (p^  q^) dv = 0.
2
spherical representation of a surface is traced out by the point m, whose coordinates are (0, 0, 1) with respect to the tri hedral T of fixed vertex. From (51) we find that the projections of a displacement of m, corresponding to a displacement along the
surface, are
(69)
The
SX=qdu + q dv,
l
&Y=
(pdu+p
2
l
dv),
&=
0.
Hence the
(70)
linear element of the spherical representation
is
da
2
= (qdu +
by
q^v)
+ (pdu+ p^dv)\
The
line defined
(65) is evidently perpendicular to the direc
tion of the displacement of m, as given by (69). Hence the tangent to the spherical representation of a curve upon a surface is perpen
dicular to the direction conjugate to the curve at the corresponding point. Therefore the tangents to a line of curvature and its rep
resentation are parallel, whereas an asymptotic direction and its representation are perpendicular ( 61).
174
72.
FUNDAMENTAL EQUATIONS
Fundamental
relations
(69)
and
we
have, for the point
and formulas. From equations on the surface,
M
(53)
=
du
(71)
""
*
^=
=i?i,
\7~
=
=
?,
= 0;
^\
and
5v
~V
7
cu
(72)
du
du
..
Consequently the following relations hold between the fundamental coefficients, the rotations, and the translations:
F= f + ^,
(73)
G=
f
in particular, the parametric system on a surface is orthog the x and y axes of the trihedral are tangent to the curves onal, and v = const, and u = const, through the vertex, equations (52) are
When,
(
74)
f=V5,
17
=
=0,
and equations (55) reduce to
(76)
r
L

Moreover, equations (45) and the similar ones for p lt q^ r t become
P
(76)
""^
T..
.
The
first
two
of equations (75) lead,
1
"
by means
1
of (76), to
sin
w
d^/~E
fo
shift\
PU
VEG
PV
which follow also from
58.
The
third of
in
remarked
equations (75) establishes the fact, previously 59, that the geodesic torsion in two orthogonal
directions differs only in sign.
FUNDAMENTAL RELATIONS
The
u
const, are represented
175
variations of the directioncosines X\, Y\, Z\ of the tangent to the curve by the motion of the point (1, 0, 0) of the trihedral T
with fixed vertex.
From
(51)
we have
5^1 cu
dZ\
55Ti
du
(78)
du
dv
cv
see that as a point describes a curve v = const. , namely the tangent to this curve undergoes an infinitesimal rotation consisting of two components, one in amount rdu about the normal to the surface and the other,
From these equations we
CM
,
qdu, about the line in the tangent plane perpendicular to the tangent to C u Consequently, by their definition, the geodesic and normal curvature of Cu are r/^/E and q/^/E respectively. Moreover, it is seen from (72) that as a point describes Cu the normal to the surface undergoes a rotation consisting of the com
.
ponents q du about the line in the tangent plane perpendicular to the tangent, and p du about the tangent. Hence, if Cu were a geodesic, the torsion would be p/VE to within the sign at least. Thus by geometrical considerations we have obtained the fundamental relations (76).
We
From
that
suppose now that the parametric system is any whatever. the definition of the differential parameters ( 37) it follows
E=
if
G=
denote functions similar to p,
.
Consequently general curve
P, $,
^
q, r,
for a
v)
= const.
and whose tangent makes the angle which passes through with the moving zaxis, we have, from (45),
M
P^VA^T cos
(79)
)
T,
&lt;
(
/^ 1\ + sm ^ cos ^1 ,
&lt;I&gt;
\ds
idco
4&gt;
r/
1\
cos
&lt;
p
cos P
,
= H. V A.cf) T sin
,
\ds
rj
sn
A 51) ff~ any other family of curves
where by
2
(III,
=
A
2
X
((,
TT)
and
ty
= const,
defines
Moreover, equations analogous to (44) are
da
_ bE
cQ
db
__
cP
aR
dc
ds
aQ
176
T.
FUNDAMENTAL EQUATIONS
.
If
now
in

da
as
da =
du
H
da dv
dv as
cu as
we
for db/ds
replace the expressions for and dc/ds, we obtain
/
l l (f&gt;(p
and
from
(47),
and similarly
Pd8=H \ A
From
du
+p
1
dv),
VA~C/&gt;
Qds=H^/~K^&gt;(qdu
(r
+ q^dv),
Eds
= 7^
du
f r^ dv).
these equations and (79)
:
we
derive the following funda
mental formulas
(
ds
j TI
= cos = sin
d&lt;&
\ as
&lt;&(p
du
+p +p
l
l
dv)
+ sin
cos
&lt;
(q
du
+
q l dv),
(80)
ds
&lt;I&gt;
(p
du
1
dv)
4&gt;
(&lt;?
c?t*
f q^ dv),
P
sin
oj
du
ds
dv
ds
p
ds
of the last of equations (80) we shall express the geodesic curvature of a curve in terms of the functions E, F, G, of their derivatives, and of the angle 6 which the curve makes
By means
with the curve v
tangent to the
= const. If we take the rraxis of the trihedral curve v = const., we obtain from the last of (80),
1
in consequence of (45),
d0
^/E du
Pg
/V G
\Pffv
dco\dv
Po~~ ds
ds
dv/ds
From
(III, 15, 16)
we
obtain
dv
If
2
EG \
dv
dv/
for p gu
dv
When
this value
and the expressions
and p gv (IV, 57) are
substituted in the
above equation, we have the formula desired:

L
__
2dvds
+
2H\du
__ E
dv
ds
EXAMPLES
1.
A necessary
is
only point in the
tangent,
be the sufficient condition that the origin of the trihedral which generates a surface to which this plane is moving zyplane
and
T
that the surface be nondevelopable.
PARALLEL SURFACES
2. Determine p so that the point of coordinates (p, describe a surface to which the xaxis of T is normal
0, 0)
;
177
with respect to
T shall
const.
examine the case when the
lines of curvature are
3.
parametric and the xaxis
is
tangent to the curve v
=
When
it is
sphere,
the parametric curves are minimal lines for both the surface and the necessary that
or
in this case the
77
=
i,
ifji
=
ii,
q
=
ip,
q\
=
ipi\
parametric curves on the surface form a conjugate system, and the
(cf.
surface
4.
is
minimal
55).
When
the asymptotic lines on a surface form an orthogonal system,
we
must have
in
^+^=
is
^
^+^=
cosw
Q&gt;
which case the surface
5.
minimal.
When
the lines of curvature are parametric, and the xaxis of
T
is
tangent
to the curve v
1
=
const., equations (80) reduce to
dw
j
T
=
sin p
/I
(
1\
)
.
sin
as
* cos
1
=
cos 2 *
H
pi
sin 2 *
,
4&gt;,
\PI
PZ/
d&lt;
p
/ q dp\
PZ
w
_
du
ds
Pz
Pi
\pi cv ds
is
p\ dPz dv\ q du ds)
s,
6.
When
cos
P2
resulting equation
the second equation in Ex. 5 is reducible to
differentiated with respect to
the
u dp
ds
sinw/ dw
p
\
2\
T/
_
2
s dp\ /du\
2
dPi/du\ 2 dv
dv \ds/ ds
ds
du \ds/
dpo
du /du\ 2
dp
/dv\ 8
7. On a surface a given curve makes the angle * with the xaxis of a trihedral T; the point of coordinates cos sin with reference to the parallel trihedral TO with fixed vertex, describes the spherical indicatrix of the tangent to the curve the directioncosines of the tangent to this curve are
P
&lt;t&gt;,
&lt;J&gt;,
;
sin
* sin w,
cos
&lt;
sin w,
cos w,
is
where w has the significance indicated in 49, and the linear element therefrom by means of (51) the second and third of formulas (80).
8.
ds/p; derive
The point #, whose coordinates with reference
sin
to
T
of Ex. 7 are
* cos w,
cos $ cos w,
sin w,
describes the spherical indicatrix of the binormal to the given curve on the surface, and its linear element is ds/r; derive therefrom the first of formulas (80).
73. Parallel surfaces.
We
inquire under
what conditions the
t
normals to a surface are normal to a second surface. In order that
this be possible, there
must
exist a function
such that the point
7",
of coordinates
(0, 0,
Q, with reference to the trihedral
describes
a surface to which the
moving
2axis is constantly normal.
Hence
178
FUNDAMENTAL EQUATIONS
8z
we must have
and consequently, by equations (51), t must may have any value whatever. We have, therefore, the theorem
0,
=
be a constant, which
:
If segments of constant
face, these segments being
other end points
is
length be laid off upon the normals
to
a sur
measured from
the surface, the locus of their
a surface with the same normals as the given surface.
These two surfaces are said to be parallel. Evidently there is an infinity of surfaces parallel to a given surface, and all of them have the same spherical representation.
Consider the surface for which
t
has the value
a,
and
call it $.
follows that the projections on the axes of (51) on S have the values placement
it
From
T of
a dis
r
f
du
(82)
=
du 77
+ ^dv f (q du + q^dv) a, jj^dv (p du + Pidv) a.
f
Comparing these
results with (53),
we
see that the displacements
on the two surfaces corresponding to the same value of dv/du are
parallel only in case equation (59) is satisfied, that
is,
when
the
point describes a line of curvature on S. But from a characteristic property of lines of curvature ( 51) it follows that the lines of curva
ture on the two surfaces correspond.
Hence we have
the theorem
:
The tangents to corresponding lines of curvature of two parallel surfaces at corresponding points are parallel.
From
first
(82)
and
(73)
we have
the following expressions for the
Y
fundamental quantities of
/S
:
y
or, in
consequence of (IV, 78),
(84)
i
/
/
PI
p
PARALLEL SURFACES
The moving
(82)
it
179
trihedral for
S can be taken
same
and thus the rotations
are the
parallel to trihedrals ; for both
T
for
,
and from
follows that the translations have the values
= + 00,
li=i+
fl
?i&gt;
^
= i?op,
*?i=&gt;?iapi
analogous to (59), (60), (66), we obtain the fundamental equations for S in terms of the functions for S. Also from (73) we have the following expressions for the
equations for
On substituting in the
second fundamental coefficients for S:
(85)
D = Da,
D = D a&,
D"
=
D"
 ag.
Since the centers of principal curvature of a surface and its are the same, it follows that parallel at corresponding points
(86)
Pi
= Pi+
a
&gt;
P2
= P2+
a
Suppose that we have a surface whose total curvature is constant and equal to 1/c 2 Evidently a sphere of radius c is of this kind, but later (Chapter VIII) it will be shown that there is a large group of surfaces with this property. We call them spherical surfaces.
.
From
so that
(86)
if
we have
take a
^_ ^
=
c,
(
^_ a = ^
)
we
we
obtain
I+li.
Pi
P*
c
Hence we have the theorem
ciated two surfaces of
of
Bonnet
*
:
With every surface of constant
total curvature
2 1/c there are asso
mean curvature
from
it.
1/ey they are parallel
to the
former and
at the distances :p c
And
conversely,
is constant and different zero there are associated two parallel surfaces, one of which has from constant total curvature and the other constant mean curvature.
With every surface whose mean curvature
M moves over a surface S the corresponding centers of principal curvature M and M describe
74. Surfaces of center.
As
a point
l
z
two surfaces S and S2 which are called the surfaces of center of S. Let C l and (72 be the lines of curvature of S through M, and D l and
1
,
7&gt;
2
the developable surfaces formed by the normals to
*Nouvelles annales de mathematiques, Ser.
1,
S along Cl
Vol. XII (1853), p. 433.
180
FUNDAMENTAL EQUATIONS
and C2 respectively. The edge of regression of D v denoted by I\, is a curve on Sl (see fig. 17), and consequently Sl is the locus of one set of evolutes of the curves Cl on S. Similarly $2 is the locus
of a set of evolutes of the curves
Cz on
is
S.
$2
are said to constitute the evolute of S,
also
Evidently any surface parallel to S
For this reason S1 and and S is their involute. an involute of S and S2
.
l
The
line
M^M^
as a generator of
Dv
is
2
tangent to I\ at
and, as a generator of
D
it is tangent to F at Hence it is a 2 z common tangent of the surfaces S and S From this it follows that the developable surface D meets S along T and envelopes Sz along a curve F 2 Its generators are con
,
.
M
Mv
z
.
l
1
{
l
.
sequently tangent to the curves conjugate to Fg ( 54). In particular, the generator
flfjJfg
is
tangent to
directions of
at Jf2 are conjugate. Similar results follow from the considera
Z&gt;
F F 2 and T 2
:
2,
and therefore the
tion of
2
.
Hence
On
the surfaces of center of a surface
S
the curves corresponding to the lines of cur
vature of
S form
a conjugate system.
Since the developable the tangent plane to $ at
FIG. 17
D
M
1
envelopes
is
*Sf , 2
2
plane at
it is
1 Z( tangent to D l all along 25), and consequently determined by M^MZ and the tangent to C[ at M. Hence the
M
plane to
is
D
l
at this point.
But
2
MM
the tangent the tangent
normal to S at
M
2
is
parallel to the tangent to
manner, the normal to
S
l
at
M
C2
at
M.
In like
l
is
parallel to the tangent to
C
l
at
M.
Thus, through each normal to S we have two perpendicular planes, of which one is tangent to one surface of center and the other to
the second surface.
But each
of these planes
is
at the
same time
tangent to one of the developables, and is the osculating plane of its edge of regression. Hence, at every point of one of these curves,
the osculating plane is perpendicular to the tangent plane to the sheet of the evolute upon which it lies, and so we have the theorem
:
The edges of regression of
normals
to
the developable surfaces formed by a surface along the lines of curvature of one family are
the
SURFACES* OF CENTER
181
geodesies on the surface of center which is the locus of these edges ; and the developable surfaces formed ly the normals along the lines of
curvature in the other family envelope this surface of center along the curves conjugate to these geodesies.
In the following sections we shall obtain, in an analytical manner, the results just deduced geometrically.
75.
Fundamental quantities
over the surface
for surfaces of center.
As
the trihe
dral
T moves
surface of center
Sr Let
S the point (0, the lines of curvature on
0, p^ describes the
S
be parametric,
and the zaxis of
T be
J.
tangent, to the curve v
const.
Now
\
/
&
L
J.
L
ri
f^
I.
A
rz
f\
so that the first
two
of equations (48)
may
be put in the form
JI, ___,,=,_
(88)
,_
__.
The projections on the moving axes of the absolute displace on S are found ment of J/J corresponding to a displacement of
M
from (51) to be
(89)
Bx l
=
0,
S^ = (rj
l
p^pj dv
= V6r
(
1
)dv,
Szj
= dp r
Hence the
(90)
linear element of
S
l
is
ds*= dri.+
/ pV Q(I^]dfi
consequently the curves p^= const, on Sl are the orthogonal tra = const., which are the edges of regression, jectories of the curves v of the developables of the normals to S along the lines of I\,
curvature v
= const.
Let us consider the moving trihedral T^ for Sl formed by the = const, and p l const, at M^ and the nor tangents to the curves v mal at this point. From (89) it follows that the first tangent has
the same direction and sense as the normal to S, and that the sec
ond tangent has the same direction as the tangent to u = const, on S, the sense being the same or different according as (1 p l /p 2 ) is
182
FUNDAMENTAL EQUATIONS
And
the normal to
positive or negative. as the tangent to v
= const,
on
,
Sl has the same direction and the contrary or same sense
accordingly.
If then
we
indicate with an accent quantities referring to the
moving
trihedral
Tv we
have
(a =c,
where
(89)
it
l
(1
=bj
is
c
=
ea,
e is
1 according as follows that
pjp^
positive or negative.
From
(92)
When
the values (91) are substituted in equations for 2\ similar
to equations (47),
we
find
Since
/
is
zero, it follows
from
v (76) that the curves
= const,
are
found geometrically. various fundamental equations for St may now be obtained of the by substituting these values in the corresponding equations
geodesies, as
The
preceding sections.
Thus, from (73) we have
which follow likewise from
(90);
and
also
Hence the parametric curves on S form a conjugate system
l
(cf.
54).
The equation
of the lines of curvature
may
be written
and the equation of the asymptotic directions
is
^^41^=0. pl$u
p? du
SURFACES OF CENTER
The expression
for
183
is
K^
the total curvature of S^
(98)
^LJj.
~du
From
(80)
the curve on
and (93) it follows that the geodesic curvature at l of Sl which makes the angle l with the curve v = const,
&lt;&
M
through M^
is
given by
Hence
is,
the radius of geodesic curvature of a curve p l = const., that is a right angle, has, in consequence of a curve for which t
&lt;J&gt;
In accordance with 57 the center of geo (87), the value p l p in the desic curvature is found by measuring off the distance p l 2 negative direction, on the 2axis of the trihedral T. Consequently
.
,
/&gt;
M
z
is this
center of curvature.
Hence we have the following theo
rem
of Beltrami:
The centers of geodesic curvature of the curves p^ = const, on St and of p 2 = const, on S., are the corresponding points on $2 and Sl
respectively.
For the sheet $2 of the evolute we
(90 d**
find the following results
:
)
= E\
du 2 +
is
the equation of the lines of curvature
(96
)
r**^
is
the equation of the asymptotic lines
^
5*
BS"
is
the expression for the total curvature
8ft
*"5FS5
dv
184
FUNDAMENTAL EQUATIONS
:
In consequence of these results we are led to the following theorems of Ribaucour,* the proof of which we leave to the reader
A necessary and sufficient condition that the lines of curvature upon
and S2 correspond is that p p 2 = c (a constant); then K^ K^ 2 1/c and the asymptotic lines upon S and $2 correspond. A necessary and sufficient condition that the asymptotic lines on Sl and S2 correspond is that there exist a functional relation between p^ and p 2
Sj
l
=
,
1
.
complementary to a given surface. We have just seen that the normals to a surface are tangent to a family of geo desies on each surface of centers. Now we prove the converse
76. Surfaces
:
The tangents
to
to
a family of geodesies on a surface
S
l
are normal
an
infinity of parallel surfaces.
Let the geodesies and their orthogonal trajectories be taken for const, and u = const, respectively, and the param the curves v
eters
chosen so that the linear element has the form
refer the surface to the trihedral formed by the tangents to the parametric curves and the normal, the zaxis being tangent to the curve v = const. Upon the latter we lay off from the point l denote the other extremity. of the surface a length X, and let
We
M
P
As
M moves over the surface the projections of the corresponding
1
displacements of
(99)
P
have the values
d\ +
du,
V +X
X
~l
=
u
dv,
 X (y,du + q,dv).
In order that the locus of P be normal to the lines J^P, we must have d\ + du = 0, and consequently
X
where
c
+
,
denotes the constant of integration whose value determines a particular one of the family of parallel surfaces. If the directioncosines of M^P with reference to fixed axes be v Yv Z^ the
X
coordinates of the surface
/S,
for
which
c=
0,
are given
by
where x^ y^
z l are
*
the coordinates of
Mr
(1872), p. 1399.
Comptes Rendus, Vol.
LXXIV
COMPLEMENTAKY SUKFACES
The
is
185
S.
surface
S
1
is
,
one of the surfaces of center of
In order
to find the other,
$2 we must determine X
so that the locus of
trihedral.
P
tangent at
P
to the zzplane of the
moving
The con
dition for this is
Hence S2
is
given by
y\
^L V
i
*!&gt;
^i
^2
^i
/
y&lt;i
aV^
dw
gVg.
dM
tfM
and the principal radii of S are expressed by
(
10
)
Pl
= u,
Pz
=udu
Bianchi*
calls
S2
the surface complementary to
S
l
for the
given
geodesic system. Beltrami has suggested the following geometrical proof of the above theorem. Of the involutes of the geodesies v const, we
consider the single infinity which meet S^ in one of the orthogonal = UQ shall prove that the locus of these curves trajectories u
.
We
is
a surface
S,
normal to the tangents to the geodesies. Consider
the tangents to the geodesies at the points of meeting of the latter with a second orthogonal trajectory u = u r The segments of these and the points P of tangents between the points of contact meeting with S are equal to one another, because they are equal
M
to the length of the geodesies between the curves moves along an Hence, as
u
lines JfP,
orthogonal trajectory l describes a second orthogonal trajectory of the latter. moves along a geodesic, describes an involute Moreover, as
M
UQ and u = u r u = u of the
P
M
P
necessarily orthogonal to MP. Since two directions on are perpendicular to JfP, the latter is normal to S.
which
is
S
EXAMPLES
1.
Obtain the results of
73 concerning parallel surfaces without making use of
the moving trihedral.
2.
Show
that the surfaces parallel to a surface of revolution are surfaces of
revolution.
*Vol.
I,
p. 293.
186
FUNDAMENTAL EQUATIONS
3. Determine the conjugate systems upon a surface such that the corresponding curves on a parallel surface form a conjugate system. 4. Determine the character of a surface S such that its asymptotic lines corre spond to conjugate lines upon a parallel surface, and find the latter surface.
5. Show that when the parametric curves are the lines of curvature of a surface, the characteristics of the 7/zplane and zzplane respectively of the moving trihe dral whose xaxis is tangent to the curve v = const, at the point are given by
(r
du
(r du
+ ri dv) y + r\ dv) x
q
(z
pi)
p%)
du
dv
pi(z
= =
0,
;
and show that these equations give the directions on the surfaces Si and S2 which are conjugate to the direction determined by dv/du.
6.
Show
that for a canal surface
(
29) one surface of centers
is
the curve of
centers of the spheres and the other
7.
is
the polar developable of this curve.
The surfaces
of center of a helicoid are helicoids of the
same axis and
parameter as the given surface.
GENERAL EXAMPLES
1.
If
t
is
an integrating factor of
^Edu\
imaginary function, then A 2 log V# is equal to the total curvature of the quadratic form E du 2 + 2 Fdudv + Gdv 2 all the functions in the latter being real.
,
v^
dv,
and
t
the conjugate
the only real surface such that its first and second fundamental quadratic forms can be the second and first forms respectively of
2.
Show
that the sphere
is
another surface.
3.
Show
that there exists a surface referred to
its lines
of curvature with the
is
linear element ds 2
=
eau (du*
+
du 2 ), where
a
is
a constant, and that the surface
developable.
4.
When
a minimal surface
is
referred to
its
minimal
lines
hence the lines of curvature and asymptotic lines can be found by quadratures.
5.
formed with respect
Establish the following identities in which the differential parameters are to the linear element
:
.
6.
Prove that
(cf.
Ex.
2, p. 1G6)
A2 *
=

4k
VJE^PI
I + }f*
VGCV\PI
( + } x(
f)
l
+
V
PZ/
\PI
GENEEAL EXAMPLES
7.
187
Show
that
z2
+
2
?/
+
z2
=
2
"FT
+
Ai TF,
(23).
the differential
8.
parameter being formed with respect to
A
necessary and
sufficient condition that all the curves of
is
an orthogonal
system on a surface be geodesies
9.
that the surface be developable.
If the geodesic
(different
from zero)
all
curvature of the curves of an orthogonal system is constant over the surface, the latter is a surface of constant negative
curvature.
10.
When
the linear element of a surface ds 2
is
in the
form
=
du
2
+
2 cos
u dudv
+
dto 2 ,
the parametric curves are said to form an equidistantial system. case the coordinates of the surface are integrals of the system
Show
that in this
du dv dy dz
dz dy
du dv
cz dx
dz dx
du dv
ex dy
_ dx
dy_
cu dv
du dv
cu dv
dv cu
cu dv
dv cu
11. If the curves v = const., u = const, form an equidistantial system, the tan the lines joining the centers of geo gents to the curves v = const, are orthogonal to desic curvature of the curves u = const, and of their orthogonal trajectories. 12.
Of
all
ment
occur
of its vertex
when
M
the displacements of a trihedral T corresponding to a small displace over the surface there are two which reduce to rotations they describes either of the lines of curvature through the point, and the
M
;
axes of rotation are situated in the planes perpendicular to the lines of curvature, each axis passing through one of the centers of principal curvature.
13.
When
a surface
3 a 2 irl d M
is
referred to
its lines
of curvature, the curves defined
by
+
3 g2
dv
duz dv
+ 3p?
dudv 2
du
+ P?
dv s
dv
=
du
in these directions at a point are possess the property that the normal sections or are superosculated by their circles of curvature (cf Ex. 9, p. 21 straight lines, These curves are called the superosculating lines of the surface. Ex.
.
;
6, p. 177).
14.
Show
Show
:
that the superosculating lines on a surface and on a parallel surface
correspond.
15.
that the Gauss equation (64) can be put in the following
form due
to
Liouville
du dv p gu du\ p gv ) where p gu and p gv denote the radii of geodesic curvature of the curves u = const, respectively.
16.
v
=
const,
and
When
may
Ex. 15
the parametric curves form an orthogonal system, the equation of be written
_!\_J:___L
VE du\ pgv )
p%u
P%V
17. Determine the surfaces which are such that one of them and a parallel divide harmonically the segment between the centers of principal curvature.
188
FUNDAMENTAL EQUATIONS
18. Determine the surfaces which are such that one of them and a parallel admit of an equivalent representation (cf. Ex. 14, p. 113) with lines of curvature
.
corresponding.
19. Derive the following properties of the surface
a2
_
ab
ft2
uv
Va 2 u b
62 v
u
(i)
+
V& 2 M+
w2
U
_
Va 2
a
&2
u Vu 2
w
;
a2
_
+
;
v
(ii) (iii)
the parametric lines are plane lines of curvature the principal radii of curvature are p\ = p% =
i&gt;,
u
algebraic of the fourth order the surfaces of center are focal conies. (iv)
the surface
is
;
20. Given a curve
C upon
gents to
M.
N
are perpendicular to C at its points at which the tangent plane to the ruled surface
S which
a surface S and the ruled surface formed by the tan the point of each generator
M
;
is
perpendicular to the tan
gent plane at
when the ruled the center of geodesic curvature of C at surface is developable, this center of geodesic curvature is the point of contact of with the edge of regression.
is
;
M to S
M
MN
21. If two surfaces have the same spherical representation of their lines of in con curvature, the locus of the point dividing the join of corresponding points is a surface with the same representation. stant ratio 22.
The
locus of the centers of geodesic curvature of a line of curvature
is
an
evolute of the latter.
23.
Show
that
when E,
is
F,
G D,
;
IX, IX of a surface are functions of a single
parameter, the surface
a helicoid, or a surface of revolution.
CHAPTER VI
SYSTEMS OF CURVES. GEODESICS
77. Asymptotic lines. We have said that the asymptotic lines on a surface are the double family of curves whose tangents at any point are determined in direction by the differential equation
D du + 2 D dudv +
2
2
D"dv
= 0.
These directions are imaginary and distinct at an elliptic point, real and distinct at a hyperbolic point, and real and coincident at a from our discussion, parabolic point. If we exclude the latter points
the asymptotic lines
neces may be taken for parametric curves. condition that they be parametric is (55) sary and sufficient
(1)
A
D = D n =Q.
(IV, 25)
Then from
we have
_ _D^_
where p
as thus
!_
denned
is
called the radius of total curvature.
)
The Codazzi equations
(V, 13
may
be written
of
which the condition of integrability
a
is
ri2i
i
d
ri2i
2
4
&lt; &gt;
ail
hail
is
r
In consequence of (V, 3) this
equivalent to
In
tives.
64
we saw
that
K
is
a function of E, F,
G and
their deriva
coefficients
Hence equations
form
(3) are
two conditions upon the
2
,
of a quadratic
(6)
E du + 2 Fdudv + G dv
2
189
190
that
it
SYSTEMS OF CURVES
may
be the linear element of a surface referred to
its
asymp
totic lines.
When
these conditions are satisfied the function
D
is
given by
(2) to within sign.
tween a surface and its follows the theorem
:
Hence, if we make no distinction be symmetric with respect to a point, from 65
A
the linear element of
its coefficients
necessary and sufficient condition that a quadratic form (6) be a surface referred to its asymptotic lines is that
satisfy equations (3);
when they are
satisfied, the
surface
is
unique.
is
For example, suppose that the total curvature of the surface every point, thus j
a2
the same at
where a
is
a constant.
In this case equations
(3)
are
cv
cu
cv
du
which, since
H
2
^ 0,
are equivalent to
dv
du
u alone, and G a function of v alone. By a suitable choice of the parameters these two functions may be given the value a2 so that the linear element of the surface can be written
Hence
E is a function
of
,
(7)
ds 2
=
a 2 (du 2
+
2 cos
o&gt;
dudv
+
dv 2),
where w denotes the angle between the asymptotic lines. Thus far the Codazzi equa tions are satisfied and only the Gauss equation (V, 12) remains to be considered. When the above values are substituted, this becomes
(8)
sinw.
dudv
to every solution of this equation there corresponds a surface of constant
Hence
curvature
whose linear element
a2
is
given by
(7).
The equation of the lines of curvature is du 2 dv 2 = 0, so that if we put 2 M!, u u f v v = 2 !, the quantities u\ and v\ are parameters of the lines of cur vature, and in terms of these the equation of the asymptotic lines is du} dv} = 0.
Hence, when either the asymptotic lines or the lines of curvature are known upon a surface of constant curvature, the other system can be found by quadratures.
the asymptotic lines are parametric, the Gauss equations (V, 7) may be written
When
^ + ^ + 5^1 =
du du
dv
0,
y ()
/OX
i
dv
72
ai
fru
l
dv~
ASYMPTOTIC LINES
where
a,
5,
191
v,
ax
,
b 1 are (5)
determinate functions of u and
da
Jo
and
in
consequence of
(10)
if
du
real linearly inde the equations
=/&gt;(!*,
Conversely,
two such equations admit three
v),
pendent integrals f^u,
&lt;*
f
z
(u, v),
f
3
(u, v),
1 ^ ) # =/2 (M, V), =/l(w V), define a surface on which the parametric curves are the asymptotic lines. For, by the elimination of a, 6, a^ b from the six equations
l
obtained by replacing 6 in
(9)
by
x, y, z
we
get
n ju 7
uy
7
v*&gt;
7
=
f\
"
= 0,
which are equivalent
As an example,
to (1), in
consequence of (IV,
2, 5).*
consider the equations
is auv + bu + cv + d, where a, b, c, rf are constants. of the choosing the fixed axes suitably, the most general form of the equations surface may be put in the form
of
which the general integral
By
From
these equations
is
it is
seen that
that the surface
a quadric.
all the asymptotic lines are straight lines, so Moreover, by the elimination of u and v from these
equations we have an equation of the form z the surface is a paraboloid.
=
ax
\
2hxy
+ by 2 + ex + dy.
Hence
78. Spherical representation of asymptotic lines.
From
(IV, 77)
we have that the
totic lines,
total
curvature of a surface, referred to
its
asymp
may
be expressed in the form
(ii)
A
ff"
=^
+
where
=
(o
o^
2
,
the linear element of the spherical represen
tation being
da 2 = (odu 2 + 2 &dudv
* Darbonx, Vol. I, p. 138. It should be noticed that the above result shows that the condition that equations (9) admit three independent integrals carries with it not only (10) but all other conditions of integrability.
192
SYSTEMS OF CURVES
this result
From
and
(2) it
follows that *
5!
#
Hence the fundamental
/
relations (IV, 74) reduce to
Jf
o\
Jf
n^/""
_
r?
"
and equations (V, 26) may be written
"
A
^
^X
P / **dX
e$X\
3X__^__ /{ \
_
du
cv
Moreover, the Codazzi equations (V, 27) are reducible to
Consider
now
the converse problem
:
To determine the condition
to be satisfied
by a parametric system
serve as the spherical
of lines on the sphere in order that they
representation of the asymptotic lines
may
on a surface.
First of
bility.
all,
Then
x,
values of
equations (15) must satisfy the condition of integrap is obtainable by a quadrature. The corresponding y, z found from equations (14) and from similar ones
are the coordinates of a surface
are parametric.
upon which the asymptotic
lines
For,
it
follows from (14) that
du
** dv dv
;
Furthermore, p
is
determined to within a constant factor
is
conse
true of x,y,z\ therefore the surface is unique quently the same to within homothetic transformations. Hence we have the following
theorem of Dini
:
A
necessary
and
sufficient condition that
a double family of curves
lines
upon the sphere be the spherical representation of the asymptotic upon a surface is that &, &, $ satisfy the equation V d ri2
the corresponding surfaces are homothetic transforms of one another,
and
* as
is
their Cartesian coordinates are
found by quadratures.
The choice
seen from
P
=
D /ft gives the surface symmetric to the one corresponding to (12),
and hence may be neglected.
(14),
FORMULAS OF LELIEUVKE
193
When equations (1) obtain, the fundamental equations (V, 28) lead to the identities
in =
121
(18)
r 11
v
2
ri2V
1
22
= r 22V_ r
=
ri2i
ri2v
rm ri2i
l2/

ri2v
i
rny \2J
r22j llJ py llJ
The
(3)
third
and fourth
of these equations are consequences also of
and
(15).
Tangential equations. In conse quence of (V, 31) equations (14) may be put in the form
79.
Formulas
of
Lelieuvre.
where
e is
1 according as the curvature of the surface
is
positive
or negative.
(20)
Hence,
if
we put
v2
^ = V/)X,
= V^epY,
vs
= ^/
*
:
we have
the following formulas due to Lelieuvre
~
du dv
_ du dv _ dudv
v* v*
Bulletin des Sciences Mathtmatiques, Vol. XII (1888), p. 126.
194
^
SYSTEMS OF CURVES
and
(15)
By means
,
of (V, 22)
we
find
from (20) that the common
Consequently
i^,
i/
is ratio of these equations i
=^

&lt;&
2,
v s are
solutions of the equation
,.
.
,
,
,.
dudv
Conversely,
\^/p dudv
:
we have
the theorem
Given three particular integrals v^
2
i&gt;
2
,
v s of
an equation of the form
(22)
d ^ = X0,
where \
is
any function whatever of u and
v ; the surface, whose co
ordinates are given by the corresponding quadratures (21), has the and the total curvature of the parametric curves for asymptotic lines,
surface
/93\
is
measured by
K~
,
to For, from (21), it is readily seen that v^ i/ 2 v z are proportional if these the directioncosines of the normal to the surface. And
directioncosines be given by (20), we are brought to (19), from which we see that the conditions (16) are satisfied.
Take, for example, the simplest case
j /
=
i
0,
\
and three solutions
//;
The coordinates
of the surface are
and similar expressions for y and
0,
/r
\
/
/
V{
0 (U)
(
Yi(V).
(I
=
1
*j &, &)
9
Q\
j
z.
When,
in particular,
we
take
(u)
=
a,w
+
&,
$i (v)
the expressions for x, y, z are of the form auv surface is a paraboloid.
+
bu
= a to + + cv +
/S,,
d,
and consequently the
the asymptotic equations (V, 22, 34) it follows that when are the tangential coordinates X, Y, Z, lines are parametric, solutions of the equations
From
W
HVd0 18^llJ du
\
I
^2/j
^22")
30
CONJUGATE SYSTEMS
EXAMPLES
1.
195
Upon a nondevelopable
surface straight lines are the only plane asymptotic
lines.
2. The asymptotic lines on a minimal surface form an orthogonal isothermal system, and their spherical images also form such a system.
3.
Show
that of all the surfaces with the linear element ds2
= du* + (u 2 + a 2
)
du2
,
one has the parametric curves for asymptotic lines and another for lines of curva ture. Determine these two surfaces.
4. The normals to a ruled surface along a generator are parallel to a plane. Prove conversely, by means of the formulas of Lelieuvre, that if the normals to a surface along the asymptotic lines in one system are parallel to a plane, which differs with the curve, the surface is ruled.
5.
If
we
take
v^
=
u,
vz
=
D,
j&gt;
3
=
0(u), the formulas of Lelieuvre define the
most general right conoid.
6. If the asymptotic lines in one system on a surface be represented on the sphere by great circles, the surface is ruled.
80. Conjugate
purpose
now
systems of parametric lines. Inversions. It is our to consider the case where the parametric lines of a
surface form a conjugate system. As thus defined, the character istics of the tangent plane, as it envelops the surface along a curve
const, at their points are the tangents to the curves u of intersection with the former curve ; and similarly for a plane const. enveloping along a curve u
v
= const.,
=
=
The
analytical condition that the parametric lines form a conju
is
(
gate system
(25)
It follows
54)
D
=0.
7) that x, y, z are solu
tions of
immediately from equations (V, an equation of the type
(26)
cudv
b are
J^ + a^ + 6^0,
du
dv
v,
where a and
functions of u and
or constants.
By
:
a
method
similar to that of
77
we prove
v) be
the converse theorem
Iffi(u, v),/2 (M, v),/3 (w,
tions of
three linearly independent real solu
an equation of
*
the type (26), the equations
(27)
=
/&gt;,*),
y=f (u,v),
z
*=f (u,v)
t
define a surface
upon which
.
the
parametric curves form a conjugate
I,
system.* 9
* Cf Darboux, Vol.
p. 122.
196
SYSTEMS OF CURVES
have seen that the lines of curvature form the only orthog onal conjugate system. Hence, in order that the parametric lines on the surface (27) be lines of curvature, we must have
We
F^fa + tyty+tete^^ du du du
dv dv dv
But
this is equivalent to the condition that
xz +yz +z 2 also be a
solution of equation (26), as is seen by substitution. have the theorem of Darboux *
:
Hence we
If
x, y, z,
# 2 {
2
?/
f
z*
are particular solutions of an equation of the
form (26), the first three serve for the rectangular coordinates of a surface, upon which the parametric lines are the lines of curvature.
Darboux
theorem
:
f has applied this result to the proof of the following
When
of the
a surface
is
face, the lines of curvature of the
latter.
transformed ly an inversion into a second sur former become lines of curvature
By
radii,
definition
is
an inversion, or a transformation by reciprocal
given by
*
where
(29)
c
denotes a constant.
(if
From
these equations
)
,
we
find that
+ f + z*) (x? + y + z, = c
z,
and by solving
for x, y,
(
,f
~
If,
"
yt+*l
now, the substitution Q
*?+**
,
__
"
?+*+?
,
be effected upon equation (26), the resulting equation in or will 4 admit, in consequence of (29) and (30), the solutions x v y^ z v c
and therefore
(31)
is
of the form
* Vol.
I,
p. 136.
t
Vol.
I, p.
207.
SURFACES OF TRANSLATION
and consequently x* + yl +
the theorem.
z? is a solution of (31),
197
Moreover, equation (26) admits unity for a particular solution,
which proves
As an example, we consider a cone of revolution. Its lines of curvature are the elements of the cone and the circular sections. When a transformation by recip
rocal radii, whose pole is any point, is applied to the cone, the transform S has two families of circles for its lines of curvature, in consequence of the above theorem
and the fact that
into circles.
circles
and straight
is
lines,
not through the pole, are transformed
the envelope of a family of spheres whose cen ters lie on its axis, and also of the oneparameter family of tangent planes the latter pass through the vertex. Since tangency is preserved in this transformation, the surface S is in two ways the envelope of a family of spheres all the spheres
Moreover, the cone
;
:
of one family pass through a point, and the centers of the spheres of the other family lie in the plane determined by the axis of the cone and the pole.
81. Surfaces of translation.
The simplest form of equation
(26) is
dudv
in
which case equations
x
,
(27) are of the type
(32)
=U
1
+r
y=
u^+V
e
=
V&gt;+r
where U^ Z7 Us are any functions whatever of u alone, and V^ F2 F3 any functions of v alone. This surface may be generated by effecting upon the curve
,
X l=
UV
Vl=
U
21=^3
a translation in which each of its points describes a curve con
gruent with the curve
*,
= F,,
y,=
r,,
Z2
.
=F
3
.
In like manner
the
it
may
curve in which each of
first
be generated by a translation of the second its points describes a curve congruent with
curve.
For
this reason the surface is called a surface of
translation. From this method of generation, as also from equa tions (32), it follows that the tangents to the curves of one family at their points of intersection with a curve of the second family are parallel to one another. Hence we have the theorem of Lie *
:
The developable enveloping a surface of translation along a gener
ating curve
is
a cylinder.
* Math. Annalen, Vol.
XIV
(1879), pp. 332367.
198
SYSTEMS OF CURVES
is
Lie has observed that the surface defined by (32)
the midpoints of the joins of points on the curves
the locus of
be that these two sets of equations define the same curve in terms of different parameters. In this case the surface is the
It
may
locus of the midpoints of all chords of the curve. These results are only a particular case of the following theorem, whose proof is
immediate
:
The
locus of the point which divides in constant ratio the joins of
points on two curves, .or all the chords of one curve, is a surface of translation ; in the latter case the curve is an asymptotic line of
the surface.
When
the equations of a surface of translation are of the form
x=U,
y
=
V,
9=Ui+V
the generators are plane curves whose planes are perpendicular. leave it to the reader to show that in this case the asymptotic lines can be found by quadratures.
We
82. Isothermalconjugate systems.
When
the asymptotic lines
upon a surface are parametric, the second quadratic form may be written X dudv. When the surface is real, so also is this quadratic
form.
Therefore, according as the curvature of the surface
is
posi
tive or negative, the parameters
u and
v are
r
conjugateimaginary
or real.
We
when
consider the former case and put
and v l are real. In terms of these parameters the second Hence the curves M = const., quadratic form is \(du+dvj). == const, form a vl conjugate system, for which
u^
t
(33)
D=
D",
D
=0.
Bianchi * has called a system of this sort isothermalconjugate. Evi an ana dently such a system bears to the second quadratic form
lytical relation similar to that of
* Vol.
an isothermalorthogonal system
107.
I, p.
ISOTHERMALCONJUGATE SYSTEMS
to the first quadratic form.
199
that
DD"
EGF* be
I)
2
&gt;
positive,
In the latter case it was only necessary and the analogous requirement, namely
by surfaces of positive curvature. Hence the theorems for isothermalorthogonal systems ( 40, 41) are translated into theorems concerning isothermalconjugate systems
0, is satisfied
all
by substituting
In particular,
Z&gt;,
IX,
D"
for E, F,
G
respectively in the formulas.
we remark
that
if
the curves
u
= const.,
all
v
= const. = const.,
on a surface form an isothermalconjugate system,
isothermalconjugate systems are given by the quantities u l and v l being defined by
other real
v1
u
= const.,
u i+
where
&lt;
il
\
() t(
u
i w)
is
is
any analytic function.
negative and
When
the curvature of the surface
we put
in the
second quadratic form \dudv,
it
becomes \(du*
dv*).
In
this case
(34)
1
D=
!&gt;",
D
=0.
Hence the curves w const, and i\ = const, form a conjugate sys tem which may be called isothermalconjugate. With each change of the parameters u and v of the asymptotic lines there is obtained a new isothermalconjugate system. Hence if u and v are parame
ters of
an isothermalconjugate system upon a surface of negative curvature, the parameters of all such systems are given by
where
&lt;
and
^r
denote arbitrary functions.
parameters for a surface are such that
It is evident that if the
(35)
D"
= ,
V
Z/=0,
where
u and v respectively, then by a which does not change the parametric curves we can reduce (35) to one of the forms (33) or (34). Hence equa tions (35) are a necessary and sufficient condition that the para metric curves form an isothermalconjugate system. Referring to
are functions of
U
and
V
change of parameters
200
77,
SYSTEMS OF CUKVES
we
see that the lines of curvature
upon a surface of constant form an isothermalconjugate system. When equation (35) is of the form (33) or (34), we say that the parameters u and v are isothermalconjugate.
total curvature
83. Spherical representation of conjugate systems.
When
the
parametric curves are conjugate, equations (IV, 69) reduce to

GI?
~~W^
FDD"
~W
.,
ED m
JiCD
these equations and (III, 15) it follows that the angle between the parametric curves on the sphere is given by
COS
,
o&gt;
From
=
&=

qp
F 
==
if
COS
O),
where the upper sign corresponds to the case of an elliptic point and the lower to a hyperbolic point. Hence we have the theorem:
The angles between two conjugate directions at a point on a sur and between the corresponding directions on the sphere, are equal
face,
or supplementary, according as the point is hyperbolic or elliptic.
curves form a conjugate system, the Codazzi equations (V, 27) reduce to
the
When
parametric
and equations (V, 26) become
(dx
D
/
X
du
dv
dX
Hence, when a system of curves upon the sphere is given, the a problem of finding the surfaces with this representation of system reduces to the solution of equations (36) and
conjugate been determined quadratures of the form (37), after X, Y, Z have of a Riccati equation. By the elimination of D by the solution
or
from equations of the second order.
D"
(36)
we
obtain a partial differential equation
CONJUGATE PARAMETRIC SYSTEMS
From the general equations (V, 28) we derive the when the parametric curves form a conjugate system
201
following, *
:
(fii\_8io S D =
li/
~w
D"
D"
rnv
~\ir i2/
f!2\
/22\
=
aiogD"
/22V ~fo~ ~i2/
(38)
/12\ =
fllV
f!2
ilJ
22
"T12/
l2/
ll
=
D /22V ~I7 ll/
!2V
The prob
84. Tangential coordinates. Pro jective transformations.
lem of finding the surfaces with a given representation of a con jugate system is treated more readily from the point of view of tangential coordinates. For, from (V, 22) and (V, 34) it is seen
that
3T,
r, Z,
and
W are particular solutions of the equation
&lt;&gt;
Hence every
A",
F,
Z
solution of this equation linearly independent of determines a surface with the given representation of a
conjugate system, and the calculation of the coordinates 2, y, z does not involve quadratures ( 67). Conversely, it is readily seen that if the tangential coordinates
satisfy
an equation of the form
d*e
ha
30 h b 00 fc#=0. 
du
dv
the coordinate lines form a conjugate system on the surface.
As an example, we determine the surfaces whose lines of curvature are repre sented on the sphere by a family of curves of ccinstant geodesic curvature and their orthogonal trajectories. If the former family be the curves v = const., and if the
linear element on the sphere be written
da 2
Edu 2 f Gdu 2 we must
,
have (IV, 60)
where
(u) is
a function of v alone.
made equal
to unity.
By a change of the parameter v this In this case equation (39) is reducible to
may
be
du
*Cf. Bianhi, Vol.
I,
p. 167.
202
The general
SYSTEMS OF CUKVES
integral of this equation
is
where and u
v
denotes a constant value of
t&gt;,
and
U and V are
arbitrary functions of u
respectively.
Hence
:
the sphere by
The determination of all the surfaces whose lines of curvature are represented on a family of curves of constant geodesic curvature and their orthogonal
two quadratures.
all
trajectories, requires
In order that among
tation of a conjugate system there
the surfaces with the same represen may be a surface for which the
system
is
conjugate,
isothermalconjugate, and the parameters be isothermalit is necessary that equations (36) be satisfied
by
iX ssiD, according
as the total curvature is positive or negative. In this case equations (36) are
01og.D_/12\
The
fllV
is
alocrT)
12V
22V
u
condition of integrability
a
rri2V
^Llim2)rdl2mi)J
When
this is satisfied
myi
may
z
17121
/22V
]
D
be found by quadratures, and then
the coordinates, by (37).
Hence we have the theorem:
A
and
the sphere represent
necessary and sufficient condition that a family of curves upon an isothermalconjugate system on a surface,
v be isothermalconjugate parameters, is that satisfy (40); then the surface is unique to within its homothetics,
,
&lt;^,
that
u and
$
and
its
coordinates are given by quadratures.
,&lt;
The following theorem concerning
directions
the invariance of conjugate
:
and asymptotic
is
lines is
to
due to Darboux
When
totic lines
a surface
subjected
a protective transformation or a
transformation by reciprocal polars, conjugate directions and asymp
are preserved.
We
When
prove this theorem geometrically.
Consider a curve
C on
a
surface
the developable circumscribing the surface along C. a projective transformation is effected upon S we obtain a
$and
D
S19 corresponding point with point to S, and C goes into a curve CjUpon S^ and D in to a developable l circumscribing Sl along
surface
D
PROJECTIVE TRANSFORMATIONS
;
203
moreover, the tangents to C and (7X correspond, as do the gener Cj ators of and r Since the generators are in each case tangent to
D
D
the curves conjugate to C and Cl respectively, the theorem is proved. In the case of a polar reciprocal transformation a plane corre
sponds to a point and vice versa, in such a way that a plane and a
point of
into
of
Sv C
D
go into a point and a plane through it. Hence S goes D1 D into Cv and the tangents to C and generators into the generators of J\ and tangents to r Hence the
it
into
,
&lt;7
theorem
is
proved.
EXAMPLES
1.
Show
that the parametric curves on the surface
2  u* + Fs u + V* where the I7 s are functions of u alone and the F s of v alone, form a conjugate system. F where U\, U are functions of 2. On the surface x U\V\, y = U V\, z u alone and FI, F2 of v alone, the parametric curves form a conjugate system and
 ^I :
+F U+V
I
_ 
C7
_
AZ
"1TTF"
Z
2
,
2
the asymptotic lines can be found
3.
(cf.
by quadratures.
form an equidistantial system
The generators
p. 187).
of a surface of translation
Ex. 10,
4.
Show
that a paraboloid
is
a surface of translation in more than one way.
is
5
.
The locus
of the midpoints of the chords of a circular helix
is
a right helicoid.
Q.
Discuss the surface of translation which
the locus of points dividing in
constant ratio the chords of a twisted cubic.
7.
From
(28) it follows that
2 dx?
_
+
,
dy?
+
a dz?
,
=
c4
(dx
(x*
2
+
dy 2
+
dz 2 )
:
+
y*
+
is
22) 2
consequently the transformation by reciprocal radii
conformal.
Determine the condition with the linear element
8.
to
9/
be satisfied by the function u so that a surface
&lt;&gt;,79,
=
a? (cos 2 w
du 2
+
that
if
shall
have the total curvature
I/a
2.
Show
the parametric curves are the
lines of curvature, they
9.
form an isothermalconjugate system.
A
necessary and sufficient condition that the linear element of a surface
referred to a conjugate system can be written
is
upon the curves on the unit sphere same
that the parametric curves be the characteristic lines. Find the condition imposed in order that they may represent these lines.
10. Conjugate systems and asymptotic lines are transformed into curves of the sort when a surface is transformed by the general protective transformation
X
=
D
ABC
y
=
D
*
=
D
Xi, y\, z\.
where A,
2&gt;,
C,
D are
linear functions of the
new coordinates
204
GEODESICS
85. Equations of geodesic lines.
We
is
have defined a geodesic to
zero at every point
is
;
be a curve whose geodesic curvature
its
conse
osculating plane at any point quently tangent plane to the surface.
perpendicular to the
follows that every geodesic an integral curve of the differential equation
From
(IV, 49)
it
upon a surface
is
(41)
, ds
.
ds/\
ds
2
ds*/
\
Y4
ds
/V
\
+
ds ds
T
7&lt;
ds)l\du
W

2 to/\ds)
Y
4i
du ds ds
fi
f 2 dv \ds
_/
\
If the
\\ds/[2 du \ds)
(\
2
i
dv ds ds
\dv
\( 2
~ Y=
du)\ds
fundamental identity
\ds/
+2+
ds ds
v, s
\ds/
which gives the relation between w, entiated with respect to s, we have
along the curve, be differ
du
d*u
d*v\
dv
d*u
d*v
L+
dv
If this
/ ,
2
424du)\ds) ds
\
dv
du) ds \dsj
4=
dv
equation and (41) be solved with respect to
72
2
72
\
and
\
T,tfu ~d*v\ F + G ^
ds
)
we
1^
obtain
,,
.
ds*/
4
F
ds
2
+F
ds

2
Y+
dv ds ds
+(
\dv
dv
2 du \ds)
2 du)\ds/
V
V=
2
cFv
(W_lMy&lt;faV
dGdu
IdG/dv \ =
2 Bv \ds)
Q
dt
If these equations
\9*
2 dv )\ds)
du ds ds

be solved with respect to
2),
and
^
we
have,
in
consequence of (V,
d*u
riii/&lt;fov
2
i
12
\
(fw
^+{ 22 V Y=o
dv
.
(42)
111 /du\*
.
rt
fl21
f 22
EQUATIONS OF GEODESICS
Every pair of solutions
v
205
of these equations of the form u =/j(), determines a geodesic on the surface, and s is its arc. =/2 (), But a geodesic may be defined in terms of u and v alone, without
s.
the introduction of the parameter
curve, then
If v
=
&lt;f&gt;(u)
defines such a
dv
du
d
*
v
/du\*
.,d*u
ct
Substituting these expressions in (42) and eliminating 2 have, to within the factor (du/ds)
,
u
&gt;
we
(43)
&lt;//
From
(42)
it
follows that
when du/ds
is
zero,
Hence, when this condition
the geodesies on a surface
;
is
not satisfied, equation (43) defines
it is satisfied,
and when
equations (43)
and
u = const,
define them.
From
exists a
,
the theory of differential equations it follows that there unique integral of (43) which takes a given value for
for
u MO and whose first derivative takes a given value Hence we have the fundamental theorem
:
u
=U
Q
.
,
Through every point on a surface there passes a unique geodesic with a given direction.
As an example, we
found
(
consider the geodesies on a surface of revolution.
We have
and
46) that the linear element of such a surface referred to its meridians parallels is of the form
(45)
ds2
=
(1
+
2
)
du 2
f
wW,
If
where z
(46)
=
(u) is the
equation of the meridian curve.
we put
and indicate the inverse of
(47)
this equation
2
&lt;Zs
by u
^(wi),
,
we have
= d^ + fdw 2
still
and the meridians and
tions (42) are
(48) (
parallels are
the parametric curves.
For
this case
equa
^i  w*y =
"
&lt;W
W
&lt;W
+
=
ds ds

206
The
first
GEODESICS
integral of the second
is
2
&lt;//
.do
ds
=
c,
where c is a constant. Eliminating ds from we have
(49)
c
this
equation and
(47),
and integrating,
f
/

=
2
w
,
_
+
C2
where Ci is a constant. The meridians Hence we have the theorem
:
v
=
const, correspond to the case c
=
0.
The geodesies upon a surface of revolution referred can be found by quadratures.
It
to its
meridians and parallels
should be remarked that equation (49) defines the geodesies upon any surface
applicable to a surface of revolution.
86. Geodesic parallels.
Geodesic parameters.
From
(43) it fol
lows that a necessary and sufficient condition that the curves v = const, on a surface be geodesies is that
parametric system be orthogonal, this condition makes it 2 be a function of u alone, say E = U By replacing necessary that
If the
E
.
I
U du
by u we do not change the parametric
lines,
and
E becomes
equal to unity.
(51)
And
G
the linear element has the form
d**=du*+Gdi?,
in general
is
where
a function of both
u and
v.
From
this it
follows that the length of the segment of a curve v the curves u = U Q and u = u^ is given by
= const, between
/!
I
ds u =
X!
I
du
V0
= u^u^
WO
Since this length is independent of v, it follows that the segments of all the geodesies v = const, included between any two orthog
onal trajectories are of equal length. In consequence of the funda mental theorem, we have that there is a unique family of geodesies which are the orthogonal trajectories of a given curve C. The above
results enable us to state the following
theorem of Gauss *
:
If geodesies be drawn orthogonal
to
a curve C,
measured upon them from
trajectory of the geodesies.
C, the locus of their
* L.C.,
p. 25.
and equal lengths be ends is an orthogonal
GEODESIC PARALLELS
207
This gives us a means of finding all the orthogonal trajectories of a family of geodesies, when one of them is known. And it sug these trajectories. Referring gests the name geodesic parallels for
to
37,
we
see that these are the curves there called parallels,
and so the theorem of
37
may
be stated thus
:
A
(52)
necessary and sufficient condition that the curves
is
= const,
&lt;f&gt;
be
geodesic parallels
that
A,* =/(*),
formed with respect to the linear any function. In order that = 0, curves measured from the curve geodesic
is
where the differential parameter
element of the surface, and
be the length of the
it
f
denotes
&lt;f&gt;
&lt;/&gt;
is
necessary
and
sufficient that
(53)
A,* = l.
(52), a
Moreover, we have seen that when a function $ satisfies new function satisfying (53) can be found by quadrature.
this function is
When
taken as
shall call
u,
the linear element has the form (51).
v geodesic parameters.
In this case
we
u and
87. Geodesic polar coordinates. The following theorem, due to Gauss,* suggests an important system of geodesic parameters:
the locus of the
If equal lengths be laid offfrom a point P on the geodesies through P, end points is an orthogonal trajectory of the geodesies.
In proving the theorem we take the geodesies for the curves and let u denote distances measured along these geo
v
= const.,
from P. The points of a curve u = const, are consequently at the same geodesic distance from P, and so we call them geodesic circles. It is our problem to show that this parametric system is
desies
orthogonal. From the choice of
it
u we know
that
E = \,
At P,
follows that
Zv
F is
dv
independent of u.
are zero.
dv
that
and hence from (50) is for u = 0, the
derivatives
for
Consequently
F
and
G
are zero
u = 0, and the former, being independent of w, is always zero. Hence the theorem is proved. We consider such a system and two points Q (u, 0), J/^w, vj on the geodesic circle of radius u. The length of the arc Q 1
M
MM
*L.c.,p.
24.
208
is
GEODESICS
/ ^Gdv. As u approaches zero the ratio Jo to the geodesies approaches the angle between the tangents at v= and v = v^ If 6 denotes this angle, we have
given by
.. a 6 = lim
v
P
=
r
I
dv.
u =o
/o
In order that v be
6, it is
necessary and sufficient that
L
:
= 1.
du Ju=
These particular geodesic coordinates are similar to polar coordi nates in the plane, and for this reason are called geodesic polar coordinates. The above results may now be stated thus
The necessary and
sufficient conditions that
a system of geodesic
coordinates be polar are
(54 )
L
J=o
=0,
L
i.
Bu
J M=0
It should be noticed, however, that it may be necessary to limit the part of the surface under consideration in order that there be a onetoone correspondence between a point and a pair of coordinates. For, it may happen that two geodesies
starting
defined
from P meet again, in which case the second point of meeting would be by two sets of coordinates.* For example, the helices are geodesies on a cylinder ( 12), and it is evident that any number of them can be made to pass through two points at a finite distance from one another by varying the angle under which they cut the elements of the cylinder. Hence, in using a system of geodesic
polar coordinates with pole at P, we consider the portion of the surface inclosed by a geodesic circle of radius r, where r is such that no two geodesies through P meet within the circle, t
When
(55)
the linear element
is
in the
form
(51), the
equation of
Gauss (V, 12) reduces to
denotes the total curvature of the surface at the pole P, which by hypothesis is not a parabolic point, from (54) and
If
Q
K
(55)
it
follows that
_
o
~
L
&lt;*
_
K
*
* Notice that the pole
is
tDarboux
also
(Vol. II, p. 408)
a singular point for such a system, because H* = for u = 0. shows that such a function r exists; this is suggested
by
94.
GEODESIC POLAK COOKDINATES
Therefore, for sufficiently small values of w,
.
209
we have
O
Hence the circumference and area
have the values *
of a geodesic circle of radius
u
=
/* 2
I
2 ITU
+
Jo
where
e
t
and
e2
denote terms of orders higher than the third and
fourth respectively.
EXAMPLES
1. 2.
Find the geodesies of an
ellipsoid of revolution.
The equations x
u,
linear element ds2
=
v (du 2
on the former are represented by parabolas on the
3.
Find the
total curvature of a surface
US
T XV*
(a 
 latter.
2
+
v define a representation of a surface with the y dv 2 ) upon the xyplane in such a way that geodesies
=
with the linear element

v 2) du?
+
2
w&gt;
dudv
a2 (
_
W2 _

+
(a
2
 u2
)
dv*
9
W2)2
where
4.
R
and a are constants and integrate the equation of geodesies for the surface.
twisted curve
is
A
a geodesic on
its
rectifying developable.
its
5. 6.
The
evolutes of a twisted curve are geodesies on
polar developable.
Along a geodesic on a surface of revolution the product of the radius of the parallel through a point and the sine of the angle of inclination of the geodesic with the meridian is constant.
7.
Upon a
Upon a
surface of revolution a curve cannot be a geodesic and loxodromic
cylindrical.
at the
8.
same time unless the surface be
helicoid the orthogonal trajectories of the helices are geodesies
and
the other geodesies can be found
by quadratures.
9. If a family of geodesies and their orthogonal trajectories on a surface form an isothermal system, the surface is applicable to a surface of revolution.
10.
The radius
varies as the cube of the distance of
of curvature of a geodesic on a cone of revolution at a point from the vertex.
P
P
88. Area of a geodesic triangle. With the aid of geodesic polar coordinates Gauss proved the following important theorem f
:
180 of the sum of the angles of a triangle formed on a surface of positive curvature, or the deficit from 180 by geodesies
The
excess over
* Bertrand, Journal de Mathematiques, Ser.
1,
Vol. XIII (1848), pp. 8086.
t L.c., p. 30.
210
of the
GEODESICS
sum of the angles of such a triangle on a surface of negative curvature, is measured by the area of the part of the sphere which
represents that triangle.
In the proof of this theorem Gauss
geodesic lines in the
made use
of the equation of
form
where 6 denotes the angle which the tangent to a geodesic at a const, through the point. This point makes with the curve v When equation is an immediate consequence of formula (V, 81). the parametric system is polar geodesic, this becomes
(57)
M =  *.
Let ABC be a triangle whose sides are geodesies, and let a, /3, 7 denote the included angles. From (IV, 7 3) it follows that the inclosed area on the sphere is given by
(58)
d=
f f// dudv =
Ipcff
dudv,
1 according as the curvature is positive or negative, where e is and the double integrals are taken over the respective areas. Let A be the pole of a polar geodesic system and AB the curve v = 0. From (55) and (58) we have
rr
 dvdu
,
o Jo Jo o
]
In consequence of (54) we have, upon integration with respect to
u,
which, by (57),
is
equivalent to
&=
For, at
v
e
f
Jo
dv
+
e
dd. Jnft
f
B
the geodesic
at
=
0,
and
C
it
BC makes the angle TT fi with the curve makes the angle 7 with the curve v = a. Hence
(7i
we have
= e(a + /3 + 7  TT),
which proves the theorem.
AREA OF A GEODESIC TRIANGLE
211
Because of the form of the second part of (58) Ci may be said to measure the total curvature of the geodesic triangle, so that the above theorem may also be stated thus
:
The
over
total
curvature of a geodesic triangle
is
equal
to the excess
180,
or deficit
from 180,
of the
sum of
the angles of the tri
is positive or negative. angle, according as the curvature
The extension
is
of these
theorems to the case of geodesic polygons
straightforward.
In the preceding discussion it has been tacitly assumed that all the points of the can be uniquely denned by polar coordinates with pole at A. We triangle shall show that this theorem is true, even if this assumption is
ABC
not made.
If the theorem is not true for ABC, it cannot be true for both of the triangles ABD and ACD obtained by joining A and the middle point of BC with a geodesic (fig. 18). For, by adding the results for the two triangles, we should have the
AD
ABC. Suppose that it is not true for A BD. Divide the latter into two triangles and apply the same reason should obtain a triangle as ing. By continuing this process we
theorem holding for
we please, inside of which a polar geodesic system would not uniquely determine each point. But a domain can be chosen about a and any other point point so that a unique geodesic passes through the given point of the domain.* Consequently the above theorem is perfectly general.
small as
By means
of the above result
we prove
the theorem
:
Tivo geodesies on a surface of negative curvature cannot meet in two points and inclose a simply connected area.
a Suppose that two geodesies through a point A pass through second point B, the two geodesies inclosing a simply connected
portion of the surface
(fig.
19).
two segments
Take any geodesic cutting these AB in points C and D. Since
the four angles
ACD, ADC, BCD,
BDC are
/D
together equal to four right angles, the sum of the angles of the two triangles ADC arid
four right angles by the sum of the angles at A and B. Therefore, in consequence of the above theorem of Gauss, the total curvature of the surface cannot be negative at all points of the area ADBC.
BDC exceed
On
the contrary,
it
curvature geodesies through a point meet again in general.
* Darboux, Vol.
II, p.
can be shown that for a surface of positive In
408
;
cf.
94.
212
fact, the
GEODESICS
exceptional points,
if
there are any,
lie in
a finite portion
which may consist of one or more simply connected parts.* For example, the geodesies on a sphere are great circles, and all of these through a point pass through the diametrically
of the surface,
Again, the helices are geodesies on a cylinder evident tnat any number of them can be made to ( 12), pass through two points at a finite distance from one another by varying the angle under A hich they cut the elements of the cyl
opposite point.
and
it is
inder.
is
Hence the domain
restricted
oi a system of polar geodesic coordinates on a surface of oositive curvature.
89. Lines of shortest length.
Geodesic curvature.
:
We
are
now
in a position to prove the
theorem
that only one geodesic passes through them, the segment of the geodesic measures the shortest dis
If two points on a surface are such
tance on the surface between the two points.
Take one
and the
of the points for the pole of a polar geodesic system 0. The coordinates of the geodesic for the curve v
=
second point are (u^ 0). The parametric equation of any other and the curve through the two points is of the form v =
^&gt;(w),
length of
its
arc
is
f
Jo
Since
tegral
G
is
&gt;
0,
the value of this in
necessarily greater than
By means
of equation (57)
we
is proved. Wj, derive another definition of geo
and the theprem
desic curvature.
Consider two points and upon a curve C, and the unique geodesies g, g tangent to C at these points (fig. 20). Let P denote the point of intersection of g and g\ and Sty the angle under which they cut. Liouville f has called Sty the angle of
M
M
geodesic contingence, because of its analogy to the ordinary angle of contingence. Now we shall prove the theorem:
The
limit of the ratio Sty/Ss, as
M
approaches M,
is the
geodesic
curvature of
C
at
M.
by H.
v.
Vol.
* For a proof of this the reader is referred to a memoir XCI (1881), pp. 2353. t Journal de Mathtmatiques, Vol. XVI (1851), p. 132.
Mangoldt, in
Crelle,
GEODESIC ELLIPSES AKD HYPERBOLAS
In the proof of this theorem we take for parametric curves the given curve (7, its geodesic parallels and their geodesic orthogonals,
the parameter u being the distance measured along the latter from C. Since the geodesic g meets the curve v = v orthogonally, the angle
under which
it
meets
v
W approaches dd given by approaches angles of the triangle M PQ approaches 18C.
Jf,
= v may
be denoted by ?r/2
y/&gt;7),
4 SO.
As
M
and the sum
S^/r
of the
Hence
approaches
dQ, so that
we have
Ss
ds
v
which
is
the expression for the geodesic curvature of the curve C.
90. Geodesic ellipses
parametric lines for a surface is
parallels.
and hyperbolas. An important system of formed by two families of geodesic
Such
a system
may
desic parallels of
two curves
be obtained by constructing the geo C^ and (72 which are not themselves
,
geodesic parallels of one another, or by taking the two families of geodesic circles with centers at any two points F^ and 2 Let u and
F
.
v
measure the geodesic distances from C^ and C2 or from
,
F
l
and
F
2
.
They must be we must have
solutions of (53).
Consequently, in terms of them,
Q.
^
EGF*~~ EGF*~
If,
as usual,
o&gt;
denotes the angle between these parametric lines,
we
have, from (III, 15, 16),
U
v= n = U
*
&gt;
sin
2
ft)
T? JF
= COSft)
.
?
sin
&lt;o
so that the linear element has the following form, due to Weingarten
/rrix
:
(59)
,9 du ds*= +
2
2 cos
.
ft)
dudv
&)
f
dv 
2
sin
a 2
Conversely,
when
the linear element
is
reducible to this form,
u and
v are solutions of (53), curves are geodesic parallels.
and consequently the parametric
v
=u
In terms of the parameters u v and v^ denned by u l i\, the linear element (59) has the form
= i^h v
l
and
(
60)
df^^aL + JuL.
sm
.
o
o
S
2
214
GEODESICS
i^
The geometrical significance of the curves of parameter is seen when the above equations are written The curves w
and
vl
are respectively the loci of the sum and difference of whose geodesic distances from C1 points and Cg, or from t and z are constant. In the latter case these
1
= const,
and
vl
= const,
F
F
,
and hyperbolas in the plane, the points Fl and F2 corresponding to the foci. For this reason they are called geodesic ellipses and hyperbolas, which names are given
curves are analogous to ellipses
likewise to the curves u^
are
=
const., v l
= const., when
.
the distances
at
measured from two curves, Cl and C2 From (60) follows once the theorem of Weingarten * A system of geodesic ellipses and hyperbolas is orthogonal.
:
By means
of (61) equation (60) can be transformed into (59), thus
proving that when the linear element of a surface is in the form (60), the parametric curves are geodesic ellipses and hyperbolas. If 6 denotes the angle which the tangent to the curve v^= const,
through a point makes with the curve v
(III, 23) that
cos u
= cos
:
ft)
i
sin
... =
= const.,
,
it
follows from
sin
Hence we have the theorem
Given any two systems of geodesic parallels upon a surface ;
corresponding geodesic ellipses included by the former.
91. Surfaces of Liouville.
the
and hyperbolas
bisect
the
angles
Dini f inquired whether there were
any surfaces with an isothermal system of geodesic ellipses and a sur hyperbolas. A necessary and sufficient condition that such
face exist
is
that the coefficients of (60) satisfy a condition of the
form (41)
^ 8in2 =
V
r
r/i
CQ8
,

,
where U^ and
denote functions of u l and
i\
respectively.
In
this case the linear
element
may
be written
V
i
*\
*Ueber die Oberfliichen fur welche einer der beiden Hauptkrummungshalbmesser eine Function des anderen ist, Crelle, Vol. LXII (1863), pp. 160173. t Annali, Ser. 2, Vol. Ill (1869), pp. 269293.
SURFACES OF LIOUVILLE
By
the change of parameters defined by
215
1
this linear
element
is
transformed into
(63)
ds*
= U+r
(
2
8)
(du* + dvt),
where
U
2
and
V
z
are functions of
u 2 and
v 2 respectively, such that
Conversely,
if
the linear element
is
in the
form
(63), it
may
be
changed
into (62)
by the transformation of coordinates
Surfaces whose linear element
first
is
reducible to the form (63) were
studied by Liouville, and on that account are called surfaces of Liouville.* To this class belong the surfaces of revolution and the
quadrics
(
96, 97).
We
may
state the above results in the
is
form
:
When
the linear element of
a surface
in the Liouville form, the
parametric curves are geodesic ellipses and hyperbolas ; these systems are the only isothermal orthogonal families of geodesic conies.^
92. Integration of the equation of geodesic lines. Having thus discussed the various properties of geodesic lines, and having seen
the advantage of knowing their equations in finite form, we return to the consideration of their differential equation and derive certain
theorems concerning
its integration. in the first place, that we know a particular first inte Suppose, gral of the general equation, that is, a family of geodesies defined
by an equation of the form
(64)
From
_2_
(IV, 58)
it
follows that
\
2
M and N must
d_
satisfy the equation
\
2
/
du \^/EN 2  2
*
t
___ FNGM
I
2
FMEN
FMN + GM / ^ \^EN  2 FMN + GM /
p. 345. p. 208, for
=
Journal de Mathematiques, Vol. XI (1846), The reader is referred to Darboux, Vol. II, under which a surface is of the Liouville type.
a discussion of the conditions
216
GEODESICS
know
that there exists a func
In consequence of this equation we
tion
&lt;f&gt;
denned by
._. (DO
dc#&gt;
)
=
======================
ENFM
d&lt;f&gt;
FNGM
==:
?
===============================
du
^EN*1FMN+GM*
A^=l.
to
V EN*2FMN+GM*
Moreover, we find that
(66)
From
(III, 31)
and
(65)
it
follows that the curves
&lt;/&gt;
= const,
are
the orthogonal trajectories of the given geodesies, and from (66) it is seen that measures distance along the geodesies from the curve $ = 0. Hence we have the theorem of Darboux *
&lt;
:
When a oneparameter family
trajectories
of geodesies
is
defined by a differ
ential equation of the first order, the finite equation of their orthogonal
can be obtained by a quadrature, which gives the geodesic
at the
parameter
same
time.
Therefore,
geodesies
is
the general first integral of the equation of known, all the geodesic parallels can be found by
when
when
is
now the converse problem of finding the geodesies the geodesic parallels are known. Suppose that we have a solution of equation (66) involving an arbitrary constant a, which
not additive.
If this equation be differentiated
quadratures. consider
We
with respect to
a,
we
get
(67)
where the
element.
the curves
differential
parameter
is
But
&lt;/&gt;
this is a necessary
formed with respect to the linear and sufficient condition ( 37) that
= const,
and the curves
(68)
^ = const.= a
da
form an orthogonal system.
are geodesies.
Hence the curves defined by
(68)
In general, this equation involves two arbitrary a and a which, as will now be shown, enter in such constants, a way that this equation gives the general integral of the differ
,
ential equation of geodesic lines.
*
Lemons, Vol.
II, p.
430;
cf.
also Bianchi, Vol.
I, p.
202.
EQUATIONS OF GEODESIC LINES
Suppose that a appears
(69)
in in equation (68),
217
:
and write the latter thus
f (u,
(67)
l
v,
=a a)
r
,
which case equation
becomes
(70)
A (*,^)=0.
direction of each of the curves (69)
is
The
ratio
given by
is
^
/
If this
be independent of a, so also by (70) Write the latter in the form
the ratio ^1
36
/dd&gt;
If this equation
and
a,
(66) be solved for
and
cu
dv
we
obtain values
independent of
so that a
would have been
additive.
Hence
/
iJL/l_, an d therefore a direction at involves a, and cu / dv If then aQ a point (MO v ) determines the value of a; call it be such that =
so also does
,
.
^.
(,
r\
*,)
4,
the geodesic v/r (%, v, passes through the point (w ?; ) and ) has the given direction at the point. Hence all the geodesies are defined by equation (68), and we have the theorem:
a
=^
,
Criven a solution of the equation
II
A
1
&lt;^
= 1,
involving an arbitrary
constant a, in such a
way
that
da
involves a; the equation
da
for
all values of a arc of the geodesies
is
the finite equation of the geodesies,
and
the
is
measured by
(/&gt;.*
By means
to Jacobi
:
of this result
we
establish the following theorem
due
If a
first integral
known, the
of the differential equation of geodesic lines be finite equation can be found by one quadrature.
integral
is
Such an
of the

form
dv
du
Cf.
= ^(u,
v, a),
Darboux, Vol.
II, p. 429.
218
GEODESICS
is
where a
(64), the
an arbitrary constant.
c/&gt;,
As
this equation is of the
form
function
defined by
=
is
P
,
(#
+
a solution of equation (66). As $ involves a in the manner the finite equation of the specified in the preceding theorem,
d(f&gt;
geodesies
is
= a.
The
surfaces of Liouville
93. Geodesies on surfaces of Liouville.
(
We
of the theorem of Jacobi. 91) afford an excellent application in the form * take the linear element
(71)
ds
2
= (U
2
V) (U?du
+
2
V?dv
),
which evidently is tion (66) becomes
no more general than
(63).
In this case equa
When
this equation is written in the
form
u*\du.
one sees that
differential equa belongs to the class of partial tions admitting an integral which is the sum of functions of u In order to obtain this integral, we put each side and v alone,
it
f
equal to a constant a and integrate.
(72)
This gives
&lt;/&gt;
= C l\ \/Ua du
of geodesies
is
f F! Va
Vdv.
Hence the equation
(73)
If 6
denotes the angle which a geodesic through a point makes
with the line v
= const,
through the point,
tan 6
it
follows from
(III,
24)
and (71) that
= y
dv
* Cf.
Darboux, Vol.
Ill, p. 9.
t
Forsyth, Differential Equations (1888),
p. 310.
SURFACES OF LIOUVILLE
If the value of
219
dv/du from equation
following
2
(73) be substituted in this
first
equation,
we
obtain the
integral of
the Gauss
equation (56):
(74)
?7sin
is
+ Fcos
2
= a.
This equation
due
to Liouville. *
EXAMPLES
1.
On
portional to the difference between the
a surface of constant curvature the area of a geodesic triangle is pro sum of the angles of the triangle and
two right angles.
2. Show that for a developable surface the be found by quadratures.
first
integral of equation (56) can
3. Given any curve C upon a surface and the developable surface which is the envelope of the tangent planes to the surface along C; show that the geodesic curvature of C is equal to the curvature of the plane curve into which C is trans formed when the developable is developed upon a plane.
4.
When
the plane
is
whose
foci are at the distance 2 c apart, the linear
referred to a system of confocal ellipses and hyperbolas element can be written
5.
A
d&lt;p
ds 2
6.
2
necessary and sufficient condition that be a perfect square.
,
be a solution of Ai0
=
1 is
that
If
1,
Ai0 =:
= did + 62 where 6\ and 62 are functions of u and v, is a solution of the curves 0i = const, are lines of length zero, and the curves B\a j 62 = const,
"
are their orthogonal trajectories.
7.
the equation Ai0
satisfies
When the linear element of a spiral surface is in the form ds 2 = e 2 (du 2 = 1 admits the solution e Z7i, where U\ is a function of
?
2
\U"
do 2 ),
M,
which
an equation of the
first
order whose integration gives thus
all
the geodesies
on the surface.
8.
For a surface with the linear element
where
(f&gt;
of v alone, the equation Ai0 = 1 admits the solution and ^ 2 requiring the solu the determination of the functions tion of a differential equation of the first order and quadratures.
V and
V\ are functions
v ( ),
u\fsi (v) f
^2
\f&gt;i
9.
If
denotes a solution of Ai0
=
1
involving a nonadditive constant a, the
linear element of the surface can be written
ca
where
(0, $} indicates the
mixed
differential
parameter
(III, 48).
*i.c.,p. 348.
220
GEODESICS
94. Lines of shortest length. Envelope of geodesies. can go a step farther than the first theorem of 89 and show that whether one or more geodesies pass through two points and 2 on a sur l face, the shortest distance on the surface between these points, if it
We
M
M
measured along one of these geodesies. Thus, =f(u) and v =f (u) define two curves C and Cl passing through the points M^ M# the parametric values of u at the points being u^ and u 2 The arc of C between these points has the length
exists, is
let v
l
.
(75)
=
v
2Fv +Gv
2
du,
where
denotes the derivative of v with respect to venience we write the above thus
:
u.
For con
(76)
s= f Jiti
we put
1
*4&gt;(ui
v,
v )du.
Furthermore,
f (u)=f(u) +
,
ea&gt;(u),
where w(u) is a function of u vanishing when u is equal to u and M 2 and e is a constant whose absolute value may be taken so small
l
that the curve
C
l
will lie in
Hence the length
of the arc
MM
1
any prescribed neighborhood of
2
C.
of
C
v
l
is
=
fc/tt
(u,
v
f
e
tw,
f
e CD
)
C?M.
Thus
j
is
a function of
e,
reducing for
it is
e
=
to
s.
Hence, in order
that the curve
C
l
be the shortest of
pass through
M
and J/2
,
the neai?by curves which necessary that the derivative of s l
all
0.
with respect to
e
be zero for
e
=
This gives
On
the assumption that
admits a continuous
first
derivative
in the interval (u^
tives,
u z ), and
continuous
&lt;f&gt;
first
and second deriva
the lefthand
member
of this equation
may
be integrated
by parts with the result
"
1
wl2
/&
\v
d , n ^lauaaO: du v
d&lt;l&gt;\
LINES OF SHORTEST LENGTH
for
o&gt;
221
&&gt;
vanishes
when u equals
u^
and u 2
.
As
the function
is
arbi
is
trary except for the above conditions upon it, this equation * equivalent to the following equation of Euler :
(77)
du
this result is applied to the particular
When
tkm
(75),
form of
(f&gt;
in equa
we have
d
F+ Gv
__
dv
I
/
_
cv
71
I
_
dv
??
"
_
~
which
readily reducible to equation (43). shortest distance between two points, if existent, measured along a geodesic through the points. This geodesic
is
Hence the
if
is is
the surface has negative total curvature at all points. unique For other surfaces more than one geo
may pass through the points if the latter are sufficiently far apart. shall now investigate the nature of this
desic
We
problem.
Let
v
f(u, a) define the family of geo
,
desies through a point J/ (w v ), and let v g (u) be the equation of their envel
=
ope
let
Cl and C2 (fig. 21), and and M,,(u v z ) denote their points of contact with the MI(UV vj is greater than Jf Jfr The envelope. Suppose that the arc 2 to distance from measured along C and ^ equal to Q
(o.
We
consider two of the geodesies
,
MM
M
l
&lt;~is
D= f
JttQ
f J^
If
is
3/2
is
considered fixed and
a.
M
l
variable, the position of the latter
determined by
The
variation of
D
with
M
1
is
given by
j da J M=MI
*
21 (Lausanne, 1744)
Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, chap, ii, cf. Bolza, Lectures on the Calculus of Variations, p. 22 (Chicago,
;
1994).
222
GEODESICS
u.,
B u t for u =
u=u
and
f = g and f = g
first
x
\
consequently the last term
f)f
is
zero.
Integrating the
u=u
(
member hy parts, and noting 26), we have
that
f is
zero for
Since C^
hence
D
the geodesies through the evolute of a curve does to a family of normals to the curve. not a geodesic, for at each point of it there Moreover, the curve
("is
a geodesic, the expression in parenthesis is zero, and does not vary with r This shows that the envelope of a point bears to them the relation which
is
M
is
is an arc connecting M^ and 2 tangent a geodesic. Hence there In this way, by taking different which is shorter than the arc of &.
M
points
M
and on & we obtain any number of arcs connecting Q which are shorter than the arc of C2 each consisting of an arc z of a geodesic such as C and the geodesic distance M^ It is then of to a point distance from Q necessarily true that the shortest
l
,
M
M
l
M M
Q
V
C2 beyond
M
within the arc
that the arc
MM M M of
Q
2
is
lies not measured along 2 However, when a domain can be chosen about (72 so small Q 2
&lt;7
.
M M
,
&lt;7
2
is
shorter than the arc
M M of
any other
curve within the domain
Another
historical
and passing through these points.*
this
problem associated with
problem
is
the following
:
t
Given an arc
length joining
C
A
and B, and
joining two points A, B on a surface ; inclosing with Co a given area.
It is evident that
to find the
curve of shortest
The area
is
given by CClfdudv.
two functions
r
&lt;
M and N can
be found in an infinity of ways such that
__
8N ~
du
dM
dv
By
the application of Green s theorem
//lldudv
where the
Since
=
we have
and is taken around the contour of the area. our problem reduces to the determination of a curve C along fixed, the which the integral C *Mdu + Ndv is constant, and whose arc AB, that is, /A
last integral is curvilinear
CQ
is
* For a more complete discussion of this problem the reader
is
referred to Darboux,
Vol. Ill, pp. 86112; Bolza, chap. v. Vol. (18.30), tin fact, it was in the solution of this problem that Minding (Crelle, function to which Bonnet (Journal de I Ecole Poly technique, p. 297) discovered the Vol. XIX (1848), p. 44) gave the name geodesic curvature..
V
ENVELOPE OF GEODESICS
integral
223
C V2?
f
2
Fv
+ Gv
^du,
is
a minimum.
From
the calculus of variations
we know that, so far as the differential equations of the solution is concerned, this as finding the curve C along which the integral is the same
problem
B
JA
is
f
VE + 2 Fv +
Gv *du
s
+ c(M + Nv
)du
is
a
minimum,
c
being a constant.
Euler
equation for this integral
d
/
F 4 GV
\
+
cto
&gt;
^
cu
+ ^.!?
dy_
_
2 f 6rv /
V E + 2 .FV f
Crt/ 2
with the formula of Bonnet (IV, 56), we see that C has con stant geodesic curvature 1/c, and c evidently depends upon the magnitude of the area between the curves. Hence we have the theorem of Minding :*
Comparing
this result
In order that a curve C joining two points shall be the shortest which, together with a given curve through these points, incloses a portion of the surface with a given area,
it is
necessary that the geodesic curvature of
C
be constant.
GENERAL EXAMPLES
1.
When
the parametric curves on the unit sphere satisfy the condition
12
1
)
a
I
12 2
)
(
12
i
J
dv

n
)
j
12
)
2
r
is
surface whose total curvature they represent the asymptotic lines on a
2.
When
the equations of the sphere have the form (III, 35), the parametric
is (1
curves are asymptotic and the equation (22)
+
wu)
2
=CU vV
20, of
which the
general integral
is
^
\f/
2
^(K) + ^(.)_
1
+
uv
where
3.
(u)
and
(v)
denote arbitrary functions.
The
sections of a surface
by
all
the planes through a fixed line
and on L, form a conjugate system.
4.
the curves of contact of the tangent cones to the surface
whose
in space, vertices are
L
Given a surface of translation x
=
u,
y
=
v,
z
=f(u) +
0(0).
Determine the
lines of
so that ( Pl + P2 )Z = const., where Z functions/ and the zaxis, and determine the angle which the normal makes with
denotes the cosine of the
curva
ture on the surface.
5.
Determine the relations between the exponents
x
m&lt;
and n

t
in the equations
= U mi V ni
,
y
= Um
*V"*
t
z
= U m3 V Hs
,
so that on the surface so defined the parametric curves shall
form a conjugate sys
can be found by quadratures. tem, and show that the asymptotic lines
*Z.c., p. 207.
224
6.
GEODESICS
The envelope
(Ui
of the family of planes
+
Fi)z
+ (Uz + V2 )y + (Us + F8 )z +
(U*
,
+ F4 =
)
0,
where the
U"
s are
functions of u alone and the
F s of
is
a surface upon which
the parametric curves are plane, and form a conjugate system.
7.
The condition
that the parametric curves form a conjugate system on the
envelope of the plane
x cos u
is
+
y sin u
f
z cot v
=/(u,
u),
that
/
be the
sum
of a function of
u alone and
of v alone
;
in this case these
curves are plane lines of curvature.
8. Find the geodesies on the surface of Ex. 7, p. 219, and determine the expres sions for the radii of curvature and torsion of a geodesic.
9. representation of two surfaces upon one another is said to be conformalconjugate when it is at the same time conformal, and every conjugate system on one surface corresponds to a conjugate system on the other. Show that the lines of
A
curvature correspond and that the characteristic lines also correspond.
10.
Given a surface of revolution z
defined by
=
ucosu, y
=
wsinw,
zf(u), and
the
function
(i)
A and c are constants a conf orjualconjugate representation of the surface is defined by MI sin I?!, z\ = upon a second surface x\ = MI cosi, y\
where
;
&lt;f&gt;(ui)
V
CUi,
C log Ui
=

du
where
F
denotes the function of M found by solving
(i)
for
&lt;
.
11. If
two families of geodesies cut under constant angle, the surface
is
developable.
12. If a surface with the linear element ds*
=
(aM
2

bv 2

c)
(du?
+
cto 2 ),
where a, 6, c are constants, is represented on the xyplane geodesies correspond to the Lissajous figures defined by
by u
= x,
v
=
y, the
where A,
Z?,
C
are constants.
13. When there is upon a surface more than one family of geodesies which, together with their orthogonal trajectories, form an isothermal system, the curva ture of the surface is constant.
14. If the principal normals of a curve meet a fixed straight line, the curve is a the case where geodesic on a surface of revolution whose axis is this line. Examine the principal normals meet the line under constant angle.
GENERAL EXAMPLES
15.
225
representation
representation of two surfaces upon one another is said to be a geodesic when to a geodesic on one surface there corresponds a geodesic on the other. Show that the representation is geodesic when points with the same
A
parametric values correspond on surfaces with the linear elements
where the IPs are functions
16.
of
u alone, the
F
s of v alone,
and h
is
a constant.
A
surface with the linear element
ds 2
=
(w*

v 4 ) [0 /itself.
where
17.
is
any function whatever, admits of a geodesic representation upon
face
necessary and sufficient condition that an orthogonal system upon a sur regarded as geodesic ellipses and hyperbolas in two ways, is that when the curves are parametric the linear element be of the Liouville form in this case
A
may be
;
these curves
may
be so regarded in an infinity of ways.
18. Of all the curves of equal length joining two points, the one which, together with a fixed curve through the points, incloses the area of greatest extent, has con
stant geodesic curvature.
draw 19. Let T be any curve upon a surface, and at two nearby points P, the geodesies g, g perpendicular to T; let C be the curve through conjugate and Q the intersection of the tangents to g to gr, P" the point where it meets g
P
P
,
and g at
the limiting position of Q, as of geodesic curvature of T at P.
P"
;
P and
P
f
approaches P,
is
the center
20.
such a
Show that if a surface S admits of geodesic representation upon a plane in way that four families of geodesies are represented by four families of par
each geodesic on the surface
is
allel lines,
p. 209).
represented by a straight line (cf Ex.
.
3,
CHAPTER
QUADRICS.
VII
RULED SURFACES. MINIMAL SURFACES
Elliptic coordinates.
95. Confocal quadrics.
Two
quadrics are
confocal
when
tions coincide.
the foci, real or imaginary, of their principal sec Hence a family of confocal quadrics is defined by
the equation
a)
where u
such that
(2)
is
A+/+A1, u u u
a
2
b
2
c
2
the parameter of the family and
a, 6, c
are constants,
a
2
&gt;
b
2
&gt;
c
2
.
For each value of
(1) defines a quadric
u, positive or negative, less
than a 2 equation
,
which
is
2
&gt;
ellipsoid
when
c
u
&gt;
oo,
&gt;
&gt;
2 u an hyperboloid of one sheet when b 2 u an Ian hyperboloid of two sheets when a
&gt;
c*,
&gt;
b
2
.
As u
zero.
approaches
Hence
the smallest axis of the ellipsoid approaches 2 the surface u = c is the portion of the zyplane,
c
2
counted twice, bounded by the
2
ellipse
(4)
a
2
c
2
b
2
2
c
2
Again, the surface u
twice,
=
b
is
the portion of the ^plane, counted
bounded by the hyperbola
which contains the center of the curve. define the focal ellipse and focal hyperbola
Equations
(4)
and
(5)
of the system.
CONFOCAL QUADEICS
Through each point
(x, y, z)
227
of the family; they are determined roots of the equation
2
in space there pass three quadries by the values of w, which are
(6)
&lt;
(u)
= (a  u) (b  u) (c  u)  x
2
2
2
y
Since
&lt;/&gt;
2
2
(a
u) (c
2
2
 u)  z
2
&lt;
2
2
(a
 u) (c  u)  u) (b  u) = 0.
2 2
(b
2
2
(a
)
&lt;
0,
c/&gt;
(b
)
&gt;
0,
(c
)
&lt;
0,
&lt;/&gt;
(,
oo)
&gt;
0,
the roots of equation (6), denoted by u l9 u the following intervals
:
,
w 3 are contained
in
(7)
a
2
&gt;
u,
&gt;
b
2
,
b
2
&gt;u,&gt;
c
2
,
c
2
&gt;
u^&gt;
oo.
seen that the surfaces corresponding to u v w 2 u s are respectively hyperboloids of two and one sheets and an ellipsoid. Fig. 22 represents three confocal quadrics; the curves on the
(3) it is
,
From
ellipsoid
are
lines
of
cur
vature, and on the hyperboloid of one sheet they are
asymptotic lines. From the definition of u v
w2 u s
,
it
follows that
&lt;
(u) is
u).
equal to
(u^u) (u 2
in (6)
is
u) (u s
When
by
is
(/&gt;
replaced
this
expression
and u
given successively the 2 2 2 values a b c we obtain *
,
,
,
FIG. 22
or
=
(8)
=
(*)(&gt;?)
These formulas express the Cartesian coordinates of a point in space in terms of the parameters of the three quadrics which
pass through the point. These parameters are called the elliptic coordinates of the point. It is evident that to each set of these
* Kirchhoff
,
Mechanik,
p. 203.
Leipsic, 1877.
228
QUABEICS
coordinates there correspond eight points in space, one in each of the eight compartments bounded by the coordinate planes.
in (8) be made constant, and the one of the parameters = others u^ %, where i Ar, be allowed to vary, these equations j
If

.
t
=
define in parametric form the surface, also defined
in (1),
1^.=
by equation which u has this constant value u r The parametric curves const, are the curves of intersection of the given const., u k
Uj
to the quadric and the double system of quadrics corresponding
parameters
If
,r\\
and uk
12
,
we put
o
7*
2
.
ng
..
the equation of the surface becomes
(10)
 + ^ + =1,
a
b
c (8)
and the parametric equations
"
reduce to
u)(a
v)
la(a
N
fb
(a
 b) (a  c)
u)(b
v)
(11)
y
~~~
\ N
 a) (b  c) (b
u) (c
(c
(b
Z ~~
\c(c
v)
b}
a}(c
Moreover, the quadrics which cut (10) in the parametric curves have the equations:
a
(12)
u
b
u
c
u
= 1,
1.
av bv cv =
In consequence of
or (11) define
(3)
and
(9)
we have
that equations (10)
(13)

an ellipsoid when a&gt;u&gt;b&gt;v&gt;c&gt;0, an hyperboloid of one sheet when a&gt;u&gt;b&gt;Q&gt;c&gt;v,
an hyperboloid of two sheets when
a&gt;0&gt;b&gt;u&gt;c&gt;v.
FUNDAMENTAL QUANTITIES
96.
229
culation
(14)
Fundamental quantities we find from (11)
*.
for central quadrics.
By
direct cal
U ( U V)
_A
^_V(VU)
/()
f(0)
where for the sake of brevity we have put
(15)
=
4 (a
 6) (b  0) (c  6).
:
We derive also the following
(ab)(a(16)
=
and
lobe
\
^
a)(c
b)
u
v
abc
u
v
(17)
JL&gt;
\
uv f(u)
N uv
f(v)
Since
ture.
F and D
And
are zero, the parametric curves are lines of curva
since the change of parameters (9) did not change the curves, we have the theorem parametric
:
The quadrics of a confocal system cut one another along
curvature,
and
the three surfaces
lines of a point cut one another through
orthogonally at the point.
This result
is
illustrated
(17)
lobe
by
fig.
22.
From
.,
(14)
and
we have
1_ ~
\abc
1
Pl
~
1
/&gt;^
NtfV
all points,
p2
N^
3 2
~wV
is
_ abc
Hence the
ellipsoid
and hyperboloid of two sheets have positive
whereas the curvature
negative at
all
curvature at
points of the hyperboloid of one sheet. If formulas (16) be written
\abc
&gt;
x
abc z
uv a
\ uv
b
uv
is
c
the distance
W from the center to the tangent plane
(19)
230
QUADEICS
:
Hence
The tangent planes to a central quadric along a curve, at points of which the total curvature of the surface is the same, are equally
distant from the center.
to the (18) we see that the umbilical points correspond The conditions (13) v. of the parameters such that u values
From
=
show that
an ellipsoid is b, and c for an hyperboloid of two sheets, whereas there are no real umbilical points for the hyperboloid of one sheet. When these
this
common
value of
u and
v for
values are substituted in (11), these points on the ellipsoid
we have
as
the coordinates of
\c(bc)
and on the hyperboloid of two sheets
It
should be noticed that these points
ellipse respectively.
lie
on the focal hyperbola
and focal
97.
Fundamental quantities
for the paraboloids.
The equation
of a paraboloid
(22)
2z
be replaced by
= ax
2
+by*
may
(23)
a:=V^,
y=V^,
z
= (au +bv
l
l ).
Hence the paraboloids
are surfaces of translation
lie in
(
81)
whose
generating curves are parabolas which
perpendicular planes.
By
direct calculation
we
find
^VS^ + ftX + l
D = 0,
//
=4
^
is
so that the equation of the lines of curvature
a dv, b
dv.
.
b
FUNDAMENTAL QUANTITIES
The general
(24)
231
integral of this equation
is
^
c is
l
,
an arbitrary constant. and v 1 in (24) are given particular values, equation (24) determines two values of c, c l and 2 in general distinct. If these
where
When u
latter values be substituted in (24) successively, we obtain in finite form the equations of the two lines of curvature through the point
(., tu.
/
\
Tf 11 c l
and
,1
cz
"U
1/11 be replaced by
+
f
A + au \

(
spectively,
we
I have, in consequence of (23), the
\
on

)
J and
A + aV
\ ov two equations
(
re
buy2 +
(25)
(1
au) x
2
= u (1 +
(1
an}
ab
f (1
av)
x2 = v
+ av)
2
ba
ab
1
When
these equations are solved for x and
y
,
we
find that
equa
tion (22) can be replaced
by
a
b
b
(26)
/
E
(27)
=
1b
~a
2"^
(1+aW +
/i
,
^
,
and the parametric curves are the lines of curvature. Now we have
a
r
b
2
(u
v)
a(a *
b)u
F = 0,
u(I+au)
b a G = ~7T5~ (U ~
a(a
b)v
4
6
2
au}(\
+ av),
b]
Vab
and
1
V [a (a
U
3 [
b) u
b] [a (a
v b)
(ab)(uv)
a (ab)ub][a(ab)vb]
(a
1
(29)
b)(u
v)
a(a
b)u
b][a(a
b)v
b]
v(l+av)
232
QLJADRICS
(27), (28),
From
(30)
and
(29)
we
obtain
W = ^Xx = [a (a
 b)u b]*[a(a  b)vb)u
b]~
[
and
=
(31)
[
a (a
a (ab)v
From
these results
we
find that the ratio
is
W/z
is
constant along
the curves for which the total curvature
constant.
We
first
suppose that
it
of (26) sign, or one
b is positive and greater than a. From the follows that u and v at a real point differ in
is equal to zero. both u and v are equal to zero.
We
consider the points at which
There are two such
points,
and
(32)
their coordinates are
,0,
Evidently these points are real only on the elliptic paraboloid. From (31) it follows that p l and p z are then equal, and conse
quently these are the umbilical points. than these u and v must differ in sign,
Since at points other we may assume that u
is always positive and v negative. Moreover, from (26) it is seen that u and v are unrestricted except in the case of the I/a. elliptic paraboloid, when v must be greater than
98. Lines of curvature
(14), (27),
and
and asymptotic 91 we have the theorem
lines on quadrics.
:
From
The
lines of curvature of a
quadric surface form an isothermal
system of the Liouville type.
Bonnet * has shown that this property is characteristic of the quadrics. There are, however, many surfaces whose lines of curva
ture form an isothermal system.
faces.
They
are called isothermic sur
The complete determination of been accomplished (cf. Ex. 3, 65).
V
all
such surfaces has never
* Meraoire sur la theorie des surfaces applicables sur une surface donne e, Journal de Ecole Poly technique, Vol. (1867), pp. 121132.
XXV
ASYMPTOTIC LINES ON QUADRICS
From
The
(17), (29),
233
and
82 follows the theorem:
lines of curvature of a quadric surface
form an
isothermal
conjugate system, and consequently
the asymptotic lines can be
found
by quadratures.
We
shall find the expressions for the coordinates in terms of
the latter in another way.
Equation (10)
is
equivalent to the pair of equations
\vS
or the pair
(34)
V&lt;y
\
Vo/
\V#
V&lt;
For each value of u equations whose points lie on the surface. And to each point on the surface there corresponds a value of u determin ing a line through the point. Hence the surface is ruled, and it is
v are
where u and
undetermined.
(33) define a line all of
nondevelopable, as seen from (18).
Again, for each value of v
equations (34) define a line whose points lie on the surface (10), and these lines are different from those of the other system.
Hence the
central quadrics are doubly ruled. necessarily the asymptotic lines. Consequently, (34) be solved for z, y, z, thus
:
These
if
lines
are
equations (33),
x
u
+
v
V^i^r+i
we have
the asymptotic lines.
vP^TT
may
y
uv
1
z
.
v
u
vP ^TT
the surface defined in terms of parameters referring to
In like manner equation (22)
^fax
or
f
be replaced by
i
i^/by
i^Tby
=
2 uz,
^/ax
^Jby Vfo/
=
==

,
u
2 vz.
V ax +
Solving these,
\
*
&gt;
V ax
i
v
we have
*
I
/
\
""""
"1
C%
tJ
c\
2uv
234
QUADKICS
in the preceding case, we see that the surface is doubly ruled,* in (36) refer to the asymptotic system of straight
As
and the parameters
lines.
Hence
:
The asymptotic
lines
on any quadric are straight
lines.
EXAMPLES
1.
The
focal conies of a family of confocal quadrics
meet the
latter in the
umbilical points.
2.
Find the characteristic
lines
on the quadrics of positive curvature.
3.
The normal
along which the total curvature
at the point.
4.
section of an ellipsoid at a point in the direction of the curve is constant is an ellipse with one of its vertices
Find the equation of the form
is
= Md
cu dv
;
(cf
.
79)
when
the corresponding
surface
5.
a hyperboloid of one sheet
when a hyperbolic
paraboloid.
Find the evolute of the hyperboloid of one sheet and derive the following
:
properties
(a)
the surface
is
algebraic of the twelfth order
;
(b) the section by a principal plane of the hyperboloid consists of a conic and the evolute of a conic
;
(c)
these sections are edges on the surface
;
(d) the
curve of intersection of the two sheets of the surface
is
cut by each of
the principal planes in four ordinary points, four double points, and four cusps, and consequently is of the twentyfourth order.
6.
Determine for the evolute of a hyperbolic paraboloid the properties analogous
5.
to those for the surface of Ex.
7.
their order
of
Deduce the equations of the surfaces parallel to a central quadric determine and the character of the sections of the surface by the principal planes the quadric find the normal curvature of the curves corresponding to the asymp
;
;
totic lines
on the quadric.
99. Geodesies on quadrics. Since the quadrics are isothermic surfaces of the Liouville type, the finite equation of the geodesies
can be found by quadratures
*
(
93).
From
(VI, 74), (14) and (27),
Moreover, the quadrics are the only doubly ruled surfaces. For consider such a sur and denote by a, b, c three of the generators in one system. A plane a through a meets 6 and c in unique points B and C and the line B(J meets a in a point A. The line ABC is a generator of the second system, and the only one of this system in the plane a. The other lines of this system meet a in the line a. On this account the plane a cuts the surface in two lines, a and ABC, that is, in a degenerate conic. Hence the surface is of the second degree.
face,
,
GEODESICS ON QUADEICS
it
235
follows that the
geodesies on any
(37)
integral of the differential equation of one of the quadrics is
first
u
is
sin
2
+ v cos =
2
a,
a constant of integration and 6 measures the angle which a geodesic, determined by a value of a, makes with the lines
where a
of curvature v
=
const.
We
We
recall that in equations (11)
and
(26)
the parameter u is greater than v, except at the umbilical points, shall discuss the general case first. where they are equal.
M
Consider a particular point (u\ v ). According as a is given the value u or v equation (37) defines the geodesic tangent at
1
f
M
u
,
1
to the line of curvature
=u
or v
v
respectively.
It is
readily seen that the other values of a, determining other geo More desies through M.\ lie in the interval between u and v
.
over, to each value of
geodesies through with respect to the directions of the lines of curvature.
this result it
M
r
a in this domain there correspond two whose tangents are symmetrically placed
From
follows also that the
and
defined by (37), when a is all the intermediate values.
whole system of geodesies is given the limiting values of u and v
We
(38)
write equation (37) in the form
(u
a) sin
2
+ (va) cos =
2
.
0,
and consider the geodesies on a central quadric defined by this Suppose, first, that a equation when a has a particular value a
is
in the
domain
v&lt;
of the values of u.
Then
at each point of these
&gt;
a have seen geodesies that these geodesies are tangent to the line of curvature u = a From (11) it follows that they lie within the zone of the surface
. 
a and consequently from (38) u
We
bounded by the two branches of the curve u = a When, now, a is positive, and con a is in the domain of the values of v, u a Hence the geodesies tangent to the sequently from (38) v a lie outside the zone bounded by the two branches of curve v
. 1
&lt;
.
the line of curvature v
Similar results are true for the parabo with the difference, as seen from (26), that the geodesies loids, a lie outside the region bounded by this curve, tangent to u whereas the curves tangent to v = a lie inside the region bounded
f .
a
by
v
=a
.
236
QUADKICS
There remains for
100. Geodesies through the umbilical points.
consideration the case where a takes the unique value which
u
so
and
v
have at the umbilical points.
a Q ) sin 2
Let
it
be denoted by
,
that the curves defined by
(39) (u
+ (v
a ) cos 2
=
:
are the umbilical geodesies.
We
have, at once, the theorem
Through each point on a quadric with real umbilical points there
pass two umbilical geodesies which are equally inclined
of curvature through the point.
to the lines
Hence two diametrically opposite umbilical points of an ellipsoid by an infinity of geodesies, and no two geodesies through the same umbilical point meet again except at the diametrically
are joined
These properties are possessed also by a family of On the great circles on a sphere through two opposite points. and on each sheet of the hyperboloid of two elliptic paraboloid sheets there are two families of umbilical geodesies, but no two
opposite point.
of the
all
same family meet except
ellipsoid (11)
at the umbilical point
common
to
curves of the family.
For the
=
b
and equations (VI, 72, 73) become
~~
a^_ _1 C
~db~
\
u
du
b
1
C
\
v
v)(v
c)
_dv
v
~4J
^\(a
u)(uc)u
4J
N(a
Similar results hold for the hyperboloid of tw sheets and the of a point P from two elliptic paraboloid. Hence the distances umbilical points (not diametrically opposite) are of the form
Hence we have
The
:
lines of curvature
are geodesic ellipses
on the quadrics with real umbilical points and hyperbolas with the umbilical points for foci.
101. Ellipsoid referred to a polar geodesic system.
A
family of
umbilical geodesies and their orthogonal trajectories constitute an excellent system for polar geodesic coordinates, because the domain is unrestricted (87) except in the case of the ellipsoid,
UMBILICAL GEODESICS
237
and then only the diametrically opposite point must be excluded. We consider such a system on the ellipsoid, and let denote the the other umbilical points (fig. 23). pole of the system and
"
,
O",
If
we put
&lt;h
i = r A
I
(40)
_
irr du  l\ 2J N(a 2J \(au)(uc) du _ 1 C u 1r
I
,
I
v
*)(
I
v
dv
~2 J \(a
it is
u)(u
c)
u
b~2jv(
a
v)(v 00
C)V
(
readily found that
1
(
j_. )(&_)
By means
(41)
of (11)
we may reduce
ds
2
the linear element to the form
= dp +
=
In order that the coordinates be polar geodesic, ^r must be replaced by another parameter measuring the angles between
the geodesies.
For the
is
ellipsoid
equation (39)
(42)
(ub)s
As
previously seen, 6 is half of one of the angles between the
two geodesies through a point M. As along approaches the geodesic joining these two
M
FIG. 23
points, the geodesic
1
O MO" ap
&&gt;,
2 approaches proaches the section # = 0. Consequently the angle or its supplementary angle. Hence the angle MOO denoted by
,
we have from
(43)
(42)
lim im
u=b, = b,r=b
?
ibV\ =
\U
b/
We
by
take
&&gt;
in place of
t/r
and indicate the relation between them
^fr =/(o&gt;).
From
(41)
we have
238
This expression
second
(44)
is
QUADRICS
satisfies
the
first
of conditions (VI, 54).
The
lim 1
j
If
2
7=^ 7^6)^1.)
u)(u
 *)
[&lt;*
[
^
(
&gt;
1
*&lt;H
1
find
c)
we make
n %
yj
du =
d&lt;p
\(a
M
^
c)u u
r
use of the formulas
b
(III, 11)
and
dv

i
=
\(a
t
\\
vd(f)
M
(40),
we
v)(v
v
b
v
j
w
v
so th.it equation (44) reduces to
lim
u=,r = b
m V(^M^)
UV
of (43)
\_\
(co(uU
g)
+ i(a^(.^i =1 V
N
J
By means
we
pass from this to
Hence the
linear element has the following form
due
to Roberts *
:
siir&&gt;
The second
1
of equations (40)
may now
1
be put in the form
2J
fjl
\l(ar
.
HI u
)(p
I
c)
u
b
b
2J \(a
&) 
fjl
HI
v)(v~
f
(7,
dv
 b) (b  c) log tan (a
\vhere
this constant,
denotes the constant of integration. In order to evaluate we consider the geodesic through the point (0, ft, 0). At this point the parameters have the values u = a, v = and the Hence the above equation may be angle co has a definite value
C
&lt;?,
o&gt;.
replaced by
i
rr
~^~
du
_i
rr
,
~^~
^
(*)&lt;*)
2
.
* Journal de Mathematiques, Vol. XIII (1848), pp. 111.
PROPERTIES OF QUADRICS
239
In like manner, for the umbilical geodesies through one of the other points (not diametrically opposite) we have
i
r
I
u
u)(u
c)
du
[
i
b
r
c
\
v
dv
v c)
2Ja \(a
u
2j
\(a
v)(v
(ab)(bc)
once from formulas these that if is any point on a line of curvature u const, or v const., we have
It follows at
M
respectively
tan


tan

= const.,
tan

cot

= const.
102. Properties of quadrics.
From
(18)
it
follows that for the
central quadrics Euler
s
equation (IV, 34) takes the form
By means
(47)
of (19)
and
(37) this reduces to
I?
R
abc
In like manner,
(48)
we have
for the paraboloids
I=
:
_J! [& +
(&_)].
Hence we have
Along a geodesic or
product
RW*
is
of curvature on a central quadric the 3 s constant, and on a paraboloid the ratio fiW /z
line
.
Consider any point P on a central quadric and a direction through P. Let a, ft, 7 be the directioncosines of the latter. The semidiameter of the ellipsoid (10) parallel to this direction is
given by
(49)
a =+
.
By
definition
240
QUADRICS
G
from
and similarly for /3 and 7. When the values of #, /, 2, E, (11) and (14) are substituted, equation (49) reduces to
1
p*
= cos
u
2 fl
sin
v
2
fl
By means
(50)
of (19)
and
(37) this
2
may
2
be reduced to
ap W
this follows the
=abc.
:
From
theorem of Joachimsthal
Along a geodesic or a line of curvature on a central quadric the to the tangent product of the semidiameter of the quadric parallel to the curve at a point P and the distance from the center to the
tangent plane at
P
is
constant.
From
(47)
and
(50)
we
obtain the equation
for all points
on the quadric.
Since
W
is
the same for all direc
tions at a point, the
correspond.
maximum and minimum Hence we have the theorem
:
values of p and
R
In
a point P principal of curvature at P.*
the central section of a quadric parallel to the tangent plane at axes are parallel to the directions of the lines the
EXAMPLES
1. On a hyperbolic paraboloid, of which the principal parabolas are equal, the locus of a point, the sum or difference of whose distances frotn the generators through the vertex of the paraboloid is constant, is a line of curvature.
2.
Find the
radii of curvature
and
torsion, at the extremity of the
mean diam
eter of an ellipsoid, of an umbilical geodesic through the pokit.
3.
on an
4.
Find the surfaces normal to the tangents to a family of umbilical geodesies 76). ellipsoid, and determine the complementary surface (cf. The geodesic distance of two diametrically opposite umbilical points on an
one half the length of the principal section through the
of the linear
ellipsoid is equal to
umbilical points.
5.
Find the form
elliptic paraboloid,
when
the parametric system
element of the hyperboloid of two sheets or the is polar geodesic with an umbilical
intersection of a geodesic through the umbilical const. , then
point for pole.
6.
If
MI and M2 are two points of
with a line of curvature v
tan

point

=
cot

=
const.
* For a
to a
more complete discussion of the geodesies on quadrics, the reader is memoir by v. Braunmuhl, in Math. Annalen, Vol. XX (1882), pp. 556686.
referred
EQUATIONS OF A RULED SURFACE
241
7. Given a line of curvature on an ellipsoid and the geodesies tangent to it; the points of intersection of pairs of these geodesies, meeting orthogonally, lie on a sphere.
8. Given the geodesies tangent to two lines of curvature ; the points of inter section of pairs of these geodesies, meeting orthogonally, lie on a sphere.
103. Equations of a ruled surface.
A
surface which can be gen
erated by the motion of a straight line is called a ruled surface. Developables are ruled surfaces for which the lines, called the
generators, are tangent to a curve. faces do not possess this property,
As
and
a general thing, ruled sur
skew surfaces. Now we make ticularly those of the skew type, limiting our discussion to the
case where the generators are real.* ruled surface is completely determined
in this case they are called a direct study of ruled surfaces, par
A
by a curve upon
the curve.
Z&gt;,
it
and the direction of the
meeting with
the directrix
__
generators at their points of
We
call the latter
and the cone formed by drawing through
FIG 24
M
a point lines parallel to the generators the directorcone. If the coordinates of a point
Q
of
D are
#
,
y^
it,
,
from a point
x
is
of
expressed in terms of the arc v measured and Z, m, n are the directioncosines of the gen
2
,
erator through Jf
(51)
the equations of the surface are
lu,
=x
If
y
= y +mu,
Q
z
= z + nu,
where u
through
the distance from
.
M
M
M
Q
to a point
M on
the generator
Q
Q
makes with the tangent
cos
denotes the angle which the generator through at Jf to then
Z&gt;,
(52)
= xJ, + y m +
r
z Q n,
where the accent indicates differentiation with respect
to v
(fig.
24).
From
(53)
(51)
we
ds*
find for the linear
2
element the expression
=du +2
cos
dudv
+ (aV +
2 bu
+ 1) dv*,
where we have put for the sake of brevity
*
We shall use the
term ruled to specify the surfaces of the skew type, and developable
for the others.
242
RULED SURFACES
directly
Since the generators are geodesies, their orthogonal trajectories arrive at this result can be found by quadratures ( 92). remarking that the equation of these trajectories is
We
by
(HI, 26)
du
is
+ cos 6
dv
=
0,
and that
a function of v alone.
shall now con Developable surfaces. sider the quantities which determine the relative positions of the generators of a ruled surface. Let g and g be two generators determined by parametric values v and v + Sv, and let X, /*, v denote the directioncosines of their common perpendicular. If the directioncosines of g and g be have I + SZ, m + 8w, n + Bn respectively, we denoted I, m, n
104. Line of striction.
We
by
;
(
l\

+ nv = 0, + \ + (m + &m) + (w + Sw) v = 0, (I + 81)
nifJL
A*
and consequently
(56)
\:fji:v
(54)
it
= (m$n
)*
n$m)
:
(n&l
l&n)
:
(ISm
mSl).
From
arid
follows that
1
(mn
by Taylor
s
 nm + (nl  ln )*+(lm  ml
l+
(56)
*
2
theorem,
(57)
= l + rftr +
may
gW+.
Hence equations
be replaced by
(58)
where
If
e t , e 2 , e3
denote expressions of the
first
and higher orders in
Sv.
Mfa
y, z)
and Jf (z+8s, y
+
%
2
+
the points of ^2) are
meeting
(fig.
of this
common
24), the length
MM
,
and g respectively perpendicular with # denoted by A, is given by
or
(60)
A=
\&x
+
/x% f
LINE OF STRICTION
From
(51), after the
243
manner
Sjc
of (57), ul
)
we
obtain
f&lt;7,
= (x
+
Bv
f
IBu
where cr involves the second and higher powers and similar values for By and Bz are substituted
(61)
of Bv.
in (60),
When
this
we have
^=p +
I I
,
where
(62)
m m
.n
1
n
of Bv.
and
(52)
(63)
e
involves
(54)
first
and higher powers
In consequence of
and
we have
/=
^.
M
In order to find
its
As
tion C,
Bv approaches zero, the point approaches a limiting posi which is called the central point of the generator. Let a
this point.
Denote the value of u for
value
we
remark that
it
follows from the equations (55) and (59) that
Sx
Bl
By
Bm
Bz
Bn
_
~
Bv Bv
If the
Sv Bv
Bv 8v
above expressions for these quantities be substituted in this equation, we have in the limit, as Bv approaches zero,
(64)
a*u
+b=
0.
Consequently
(65)
The
locus of the central points
is
called the line of striction.
Its
is a necessary and parametric equation is (64). Evidently b sufficient condition that the line of striction be the directrix.
From
(61)
is
and
generators
(66)
(63) it is seen that the distance of the second order when
between nearby
a2
loss
sm2
6&gt;
6
2
=:0.
line of striction
Without
of generality for directrix,, in which case
we may take the we may have sin#
=:0, that
is,
the
244
KULED SURFACES
generators are tangent to the directrix. Another possibility is afforded by a 0. From (54) it is seen that the only real sur
faces satisfying this condition are cylinders.
Hence
(cf.
4)
:
necessary and sufficient condition that a ruled surface, other than a cylinder, be developable is that the distance between nearby genera tors be of the second or higher orders ; in this case the edge of regres
sion is the line of striction.
A
105. Central plane.
Parameter of distribution.
plane
to a ruled surface at a point
It
M necessarily
(
The tangent
contains the
generator through M.
has been found
is
opable surface this plane
tangent at
We
25) that for a devel points of the generator. shall see that in the case of skew
all
surfaces the tangent plane varies as deter moves along the generator.
M
We
mine the character
finding
the
at
plane
M
by which the tangent angle makes with the tangent
of this variation
plane at the central point
C
of the
gen
erator through M. The tangent plane at C is called the central plane.
Let g and g l be two generators, and
of g draw the plane pendicular (fig. 25). Through the point normal to g it meets g^ in and the line through parallel v and The limiting positions of the planes to ff l in 2
;
MM M
their
common
per
M
M
M
(
.
M MM, as g^ approaches
,
and at C, the tangent planes at the limiting position of M. The angle between these planes, de is equal to MMJtt^ and the angle between g and g v noted by
g, are
M
M^MM
denoted by
cr,
is
MMM
equal to
MMM
2
.
By
construction
MM M
2
l
and
2
are right angles.
Hence
= MM. =
tan
.,
d&gt;
=
2
MM ton
a
MM
(7,
In the limit
M
is
the central point
v
tan&lt;f&gt;=lim
,,
tan&lt;f&gt;
(u
2
and so we have
^ pdv
f
a)da
=
,
(u
a)a
m
p
&i 2 )
for
we have
da* = lim (SI
+ 8m
= a W.
PARAMETER OF DISTRIBUTION
It is
245
customary to write the above equation in the form
(67)
The function
It
is
ft thus defined is called the parameter of distribution. the limit of the ratio of the shortest distance between two
generators and their included angle. parameter u, we have the theorem
:
As
it is
independent of the
The tangent of
surface at a point
distance of
the angle between the tangent plane to a ruled
M and
the
the central
M from
plane
is
proportional
to
the
central point.
From
to
this it follows that as
M moves along a generator from
Hence the tangent planes
oo
+ oo,
varies from
(/&gt;
Tr/2 to 7r/2.
at
the infinitely distant points are perpendicular to the central plane. Since /3=0 is the condition that a surface be developable, the
tangent plane
is
the same at
all
points of the generator.
We shall now derive equation (67) analytically.
)
From (51) we find
that the directioncosines of the normal to the surface are of the form
(
X=(mz
2
nyQ ) + (mn
m n) u
^
to
;
the
expressions
for
,
directioncosines JT
Y and Z are similar F Z of the normal at
,
the above.
The
Q
the central point are
this
obtained from these by replacing u by
/\Q\
r\o
x/\
a.
From
we have
2
^T
~V~V
_ 2 (mz[
which leads to
ny ^f + 2 (mz[
2
ny Q } (mn
)*
(aV + 2 bu + sin
(a
V+
m n} (u + a) f a ua
2 ba
+ sin
2
)*
^2J
and
a
2
_.
a (u
4/^2 a)
From
(70) (
this equation
(67)
we have
/
._ ^~
is
_
its
"
2
I
m
n
When
the surface
defined by
linear element,
@
is
thus deter
mined only
that this
of the
is
to within an algebraic sign.
We
is
shall find,
however,
not the case
(51).
when
the surface
defined by equations
form
246
EJJLED SURFACES
end we take a particular generator g for the for g we have
this
zaxis.
To Then
Let also the central plane be taken for the zzplane and the central = 0. Since the point for the origin. From (68) it follows that y Q b= and consequently I = 0. Hence origin is the central point,
the equation of the tangent plane at a point of g has the simple form
(71)
m u%
XO T)
=
0,
f and rj being current coordinates. If the coordinate axes have the usual orientation, and the angle * is measured positively in the direction from the positive #axis to the positive ^/axis,
from equation
(7 2)
(71)
we have
tan
*
= mu
.
Comparing
values,
with equation (67), we find for ft the value xJm In order to obtain the same value from (70) for these particular
this
we must
take
the negative sign.
\
Hence we have,
in
general,
(73)
/
/
= i
2
I
I
m m
1
n n
from (72) that, as a point moves along a generator in the direction of u increasing, the motion of the tangent plane is that of a righthanded or lefthanded screw, according as ft is
It is seen
negative or positive.
EXAMPLES
1
.
Show
that for the ruled surface denned by
2J
2
,.
y
_ =
i
r
.
^
.
,
.
*
~2&gt;
Cu&lt;t&gt;du
where and are any functions of w, the directrix and the generators are minimal. Determine under what condition the curvature of the surface is constant.
\f/
2.
Determine the condition that the directrix of a ruled surface be a geodesic.
PARAMETER OF DISTRIBUTION
3. Prove, by are defined by
247
means
of (62), that the lines of curvature of a surface F(x, y, z)
=
^
dx
dy,
d_F
dz
dF
dx
&gt;*,
cF
dz
dy
a**,
dy
*?*
dz
4. The right helicoid is the only ruled surface whose generators are the principal normals of their orthogonal trajectories. Find the parameter of distribution.
5
.
Prove for the hyperboloid of revolution of one sheet that
(a)
(6)
:
the
minimum
circle is the line of striction
is
and a geodesic
;
the parameter of distribution
constant.
6. With every point P on a ruled surface there is associated another point P on the same generator, such that the tangent planes at these points are perpendicular. Prove that the product OP OP where denotes the central point, has the same value for all points P on the same generator.
,
7.
8.
The normals The
is
to
a ruled surface along a generator form a hyperbolic paraboloid.
erator
9.
crossratio of four tangent planes to a ruled surface at points of a gen equal to the crossratio of the points.
two ruled surfaces are symmetric with respect to a plane, the values of the parameter of distribution for homologous generators differ only in sign.
If
106. Particular
form
of the linear element.
erties of ruled surfaces are readily obtained
is
given a particular form, which we will Let an orthogonal trajectory of the generators be taken for the directrix. In this case
(74)
If
A number of prop when the linear element now deduce.
*,=,
u
f,*.
= u,
vl
we make
the change of parameters,
(75)
= C a dv,
I
v
Jo
the linear element (53)
(76)
ds*
is
reducible to
2 2
= du* + [(u  a) + /3
]
dv*.
is
The angle 6 which
given by
(77)
a curve v l =f(u)
makes with the generators
tan0
= V(w
is
Also the expression for the total curvature
(78)
JT =

f*
248
KULED SURFACES
=
Hence a real ruled surface has no elliptic points. All the points are hyperbolic except along the generators for which /3 0, and at the infinitely distant points on each generator. Consequently
the linear element of a developable surface
(79)
ds*
may
be put in the form
=
du"
+ (u  a)
2
dv*.
face the latter has the
Also, in the region of the infinitely distant points of a ruled sur character of a developable surface. As
another consequence of (78) we have that, for the points of a generator the curvature is greatest in absolute value at the cen
tral point,
and that
at points equally distant
from the
latter it
has the same value.
When
the linear element
is
in the
form
(76), the
Gauss equation
of geodesies (VI, 56) has the form
V(M
 a) + @*d6 + (ua) dv = 0.
2
1
An
immediate consequence
is
the theorem of Bonnet
:
If a curve upon a ruled surface has two of the following properties, it has the third also, namely that it cut the generators under constant striction. angle, that it be a geodesic and that it be the line of
formed by the family of straight lines which cut a twisted curve under constant angle and are perpen
surface of this kind
is
A
dicular to
principal normals. formed of the binomials of a curve.
its
A
particular case
It is readily
is
the surface
(73)
shown from
is
that the parameter of distribution of this surface radius of torsion of the curve.
,
equal to the
107. Asymptotic lines. Orthogonal parametric systems. The erators are necessarily asymptotic lines on a ruled surface.
gen
We
(68)
consider
now
the other family of these lines.
r
From
(51)
and
we
find
i&gt;
(80)
= 0, D
1 =H
I
m
n
"
m
z"+n"u
n
z
+nu
Hence the
differential
of asymp equation of the other family
totic lines is of the
form
dv
ASYMPTOTIC LINES
where
,
249
Jf,
N are
we
functions of
Riccati t}^pe,
have, from
As this is an equation of the the theorem of Serret: 14,
v.
The four points in which each generator of a ruled surface by four curved asymptotic lines are in constant crossratio.
is
cut
From
lines
is
14
it
follows also that
when one
of these asymptotic
known
the surface
(76),
the others can be found by quadratures.
is
When
in the
(81)
referred to an orthogonal system and the linear element
ds 2
is
form
written
=
du 2
+
a 2 [(u

a)
2
+
2
/3 ]
dv 2
,
the expressions (80) can be given a simpler form.
From
(73)
and
(81)
we have
From
and
the equations
Lx ol
by
= = =
0,
Sz6 2
=
1,
2Z 2
=
1,
(54)
we
obtain,
differentiation,
0,
Zzfceo
Ll
l"
ZM =
,
0,
Zatf
J
aa
ZK"
where
t
is
defined by
D"
zr
6
==
t.
a2
,
ZJ x6
= b, = & ,
If the
expression for
in (80) be multiplied
hand member
of (73), and the result be divided consequence of the above identities,
D"
by the determinant of the righta 2 /3, we have, in by its equal,
=
i
2
[w (to?
2

aa
fc)
+
u
(2 tb

aa

66
)
+t0,
&
].
If equations (74) be solved for a and 6 as functions of expressions be substituted in this equation, we have
D"
a and
and the resulting
= ~{r[(u 
a)
2
+ ^] + ?(u 
a)
+ /3a },
by
(75),
is
where the primes indicate differentiation with respect / defined by
to Vi, given
and r
From
(82)
the above equations
it
follows that the
mean curvature
(cf
.
52) is express
ible in the
form
J

+
*
=
r

^u ~
a)2
+
^+
2 a)
Pi
Pz
[(u
EXAMPLES
1.
When
the linear element of a ruled surface
is
in the
form
(76), the direction
cosines of the limiting position of the
?&lt;L?,
/3
common
perpendicular to two generators are
z
V*
+
&lt;
+n
o/3
t
aft
250
MINIMAL SURFACES
with real 2. Prove that the developable surfaces are the only ruled surfaces generators whose total curvature is constant.
3.
Show
x
_ u ^y
4.
totic lines
that the perpendicular upon the zaxis from any point of the cubic w a lies i n the osculating plane at the point, and lind the asymp M2 z on the ruled surface generated by this perpendicular.
?
Determine the function
x
in the equations
w,
=
y
=
un
,
z
=
0(u),
so that the osculating plane at
any point M of
this curve shall pass
lines
projection ated by the line
5.
P of
M on the yaxis.
MP.
Find the asymptotic
through the on the surface gener
Show
6
that the equations
where
z = ufwcos0, x = M sin cos ^, y = wsinflsin^, define the most general ruled surface with a rec and ^ are functions of lines can be integrated tilinear directrix, and prove that the equation of asymptotic two quadratures. Discuss the case where is constant. by ruled surface the following are 6. Concerning the curved asymptotic lines on a
t&gt;,
to be proved
if
:
one of them is an orthogonal trajectory of the generators, the determina (a) tion of the rest reduces to quadratures are curves of Bertrand if two of them are orthogonal trajectories, they (6) surface is a right helicoid. if all of them are orthogonal trajectories, the (c) be an asymptotic line, and 7. Determine the condition that the line of striction
;
;
show that
8.
in this case the other
curved asymptotic lines can be found by quadratures.
is generated by a line pass = z = 0, + yfz = l. Show that these lines through the two lines x = 0, y ing line of striction. and the line x = 0, x + y + z = 1 are double lines. Find the
Find a ruled surface of the fourth degree which
;
of whose lines of curvature 9. The right helicoid is the only ruled surface each on any other ruled surface cuts the generators under constant angle however, this property. there are in general four lines of curvature which have
;
In 1760 Lagrange extended to double theorems about simple integrals in the calculus integrals the Euler * of variations, and as an example he proposed the following problem
108.
Minimal surfaces.
:
Given a closed curve C and a connected surface S bounded by
curve;
to
the
determine
8
so that the inclosed area shall be a
minimum.
If the surface
be denned by the equation
z
=f(x,
y),
that the inte the problem requires the determination of f(x, y) so
gral
(cf.
Ex.
1, p.
77)
* CEuvres de
Lagrange, Vol.
I,
pp. 354357.
Paris, 1867.
MINIMAL SURFACES
minimum. As shown by Lagrange, the condition
(83)
251
extended over the portion of the surface bounded by C shall be a
for this is
or, in
other form,
(1
(84)
+ q )r  2pqs + (1 + p*)t = 0.
z
Lagrange
left
the
solution
of
the
Meusnier,*
sixteen
years
later,
problem in this form, and proved that this equation is
equivalent to the vanishing of the mean curvature ( 52), thus showing that the surfaces furnishing the solution of Lagrange s problem are characterized by the geometrical property which now
is
name
usually taken as the definition of minimal surfaces; however, the indicates the connection with the definition of Lagrange. f
In what follows we purpose giving a discussion of minimal sur faces from the standpoint of their definition as the surfaces whose
mean curvature is zero at all the points. At each point of such a surface the principal radii differ only in sign, and so every point is a hyperbolic point and its Dupin indicatrix is an equilateral hyperbola. Consequently minimal surfaces are characterized by
the property that their asymptotic lines form an orthogonal sys tem. Moreover, the tangents to the two asymptotic lines at a
point bisect the angles between the lines of curvature at the point,
and vice
versa.
We recall the formulas giving the relations between the funda mental quantities of a surface and its spherical representation
(IV, 70)
(85)
:
(o
=
we have
at once the
From
these
theorem
:
The necessary and
sufficient condition that the spherical
represen
tation of a surface be conformal is that the surface be minimal.
*
Memoire sur
la
courbure des surfaces, Memoires des Savants Strangers, Vol.
X
(1785), p. 477.
t For a historical sketch of the development of the theory of minimal surfaces and a complete discussion of them the reader is referred to the Lemons of Darboux (Vol. I, pp. 267 et seq.). The questions in the calculus of variations involved in the study of mini
mal surfaces are treated by Riemann, Gesammelte Werke, p. 287 (Leipzig, Schwarz, Gesammelte Abhandlungen, Vol. I, pp. 223, 270 (Berlin, 1890).
1876)
;
and by
252
MINIMAL SURFACES
Hence isothermal orthogonal systems on the surface are repre sented by similar systems on the sphere, and conversely. All the isothermal orthogonal systems on the sphere are known ( 35, 40).
Suppose that one of these systems element is *
is
parametric and that the linear
From
(86)
it
the general condition for minimal surfaces (IV, 77), namely
&lt;D"
+ 3D  2 &D =
f
0,
follows that in this case
n
1
In consequence of this the Codazzi equations (V, 27) are reducible to
(87)
?^0, dv du
D or D
we
~IQ
f + ~? =
du
dv

By
eliminating of the equation
find that both
D
and
D
1
are integrals
a?
Hence the most general form
(88)
+
of
^Q = ^?
*
D
is
D = $ (u + iv) + ^(u iv),
and
&lt;f&gt;
where
(89)
i/r
are arbitrary functions.
Then from
(87)
we have
D=
~D"
=
i((j)
ty+c,
To each pair of functions c is the constant of integration. there corresponds a minimal surface whose Cartesian coordi T/T nates are given by the quadratures (V, 26), namely
where
&lt;,
(
90)
=du
\\
du
dv
dv
\
du
dv
and similar expressions in y and z. Evidently the surface is real and i/r are conjugate functions. only when In obtaining the preceding results we have tacitly assumed that neither D nor D is zero. We notice that either may be zero and
&lt;f&gt;
1
then the other
is a constant, which These results may be stated thus:
is
zero only for the plane.
Every isothermal system on
lines of curvature of
lines of another
a unique
the sphere is the representation of the minimal surface and of the asymptotic
minimal surface.
LINES OF CURVATURE
The converse
also is true,
AND ASYMPTOTIC LINES
253
namely:
the
The spherical representations of the lines of curvature and of asymptotic lines of a minimal surface are isothermal systems.
For,
if
the lines of curvature are parametric, equation (86)
may
be replaced by
D = p^
D
D = _ pg
&gt;,
where p
sign.
equal to either principal radius to within its algebraic = = are substituted in the When these values and
is
&
Codazzi equations (V, 27), we obtain
so that
/g=*U/V, which proves the
to
first
part of the theorem
(
41).
When the asymptotic lines are parametric, we have Z&gt;=D"=c^=0,
and equations (V, 27) reduce
cu
(&gt;!")=
from which
it
follows that
&lt;~/^=
U/V.
lines. Adjoint minimal return to the consideration of equations (87) and investigate first the minimal surface with its lines of curvature represented by an isothermal system. Without loss of generality,*
109. Lines of curvature
and asymptotic
surfaces.
We
we may
(91)
take
D=
(IV, 77)
it
D"
= 1,
&gt;
=0.
From
follows that
PiP 2
where
=  = ~X Pi = ^ = p
2
2
,
/&gt;
E=G =
2 J.
p,
Hence we have
the theorem
:
The parameters of
ical representation
the lines of curvature of
a minimal surface
may
be so chosen that the linear elements of the surface
and of
its
spher
have the respective forms
2
2
ds
2
= p (du + dv
),
dd 2 =  (du 2 + dv 2 ),
P
where p
is the *
absolute value of each principal radius.
other value of the constant leads to homothetic surfaces.
Any
254
In like manner
(92)
MINIMAL SURFACES
we may
take, for the solution of equations (87),
I&gt;"=Q,
D=
find
..
D = l.
E=G = p,
:
Again we
,
J = PiP*
= \\
Pi
so that
we have
a result similar to the above
The parameters of the asymptotic
lines of a
minimal surface may
be so chosen that the linear elements of the surface
and of
2
its
spherical
representation have the respective forms
ds*
= p (du* +
di?),
d&lt;r*
=
(du*
+ dv
),
where p
is
the absolute value of each principal radius.
the symmetric form of equations (87) it follows that if one set of solutions, another set is given by (88) and (89) represent
From
These values are such that
which
is
respond
the condition that asymptotic lines on either surface cor to a conjugate system on the other ( 56). When this
is
condition
satisfied
by two minimal
surfaces,
and the tangent
are parallel, the two surfaces are planes at corresponding points said to be the adjoints of one another. Hence a pair of functions
&lt;,
determines a pair of adjoint minimal surfaces. When, in par surface a*e parametric, the ticular, the asymptotic lines on one and on the other the values (91). functions have the values
&gt;/r
(92),
It follows, then,
from
(90), that
its
between the Cartesian coordinates
of a
minimal surface and
cjx\_
relations hold: adjoint the following
foi
_dx
dv
/
=
fa.
cu
dv
s
du
z s,
and
similar expressions in the
and
when
z).
the parametric
curves are asymptotic on the locus of (#, #, 110. Minimal curves on a minimal surface.
The
lines of length
When zero upon a minimal surface are of fundamental importance. of the surface the equations they are taken for parametric curves, take a simple form, which we shall now obtain.
MINIMAL CURVES
we have
(94)
255
Since the lines of length zero, or minimal lines, are parametric,
^ = = 0.
(85)
it
follows that the parametric lines on the sphere also are minimal lines, that is, the imaginary rectilinear generators. And from (86) we find that 1) is zero. Conversely, when the latter is
zero,
From
and the parametric
lines are
minimal curves,
it
follows from
(IV, 33) that
Km
is
equal to zero.
Hence
:
A
necessary and sufficient condition that a surface be minimal
is
that the lines of length zero
form a conjugate system.*
In consequence of (94) and (VI, 26) the point equation of a minimal surface, referred to its minimal lines, is
ducv
Hence the
finite
equations of the surface are of the form
where U^
T/2 ,
U
s
are functions
of
u alone, and
F
x,
F
2,
F
3
are
functions of v alone, satisfying the conditions
(96)
U? + V? + U? =
(95)
(
2
0,
F{
+ Fi +
a
2
Fj
=
0.
From
lation
it is
seen that minimal surfaces are surfaces of trans
(96)
81),
and from
that
the
generators
are
minimal
81
curves
(
22).
In consequence of the second theorem of
:
we
may
state this result thus
minimal surface is the locus of points on two minimal curves.
In
A
the midpoints of the joins of
22 we found that the Cartesian coordinates of any minimal curve are expressible in the form
(97)
f (1  u*)F(u) du,
i
f (1 + u
2 )
F(u) du,
2
Cu F(u) du.
is
*This follows also from the fact that an equilateral hyperbola which the directions with angular coefficients i are conjugate.
the only conic for
256
MINIMAL SURFACES
Hence by the above theorem the following equations, due to Enneper *, define a minimal surface referred to its minimal lines
:
(98)
z=
4&gt;
I
u F(u) du
+
I
v&(v) dv,
are any analytic functions whatever. Moreover, where F and minimal surface can be defined by equations of this form. any For, the only apparent lack of generality is due to the fact that
the algebraic signs of the expressions (98) are not determined by equations (96), and consequently the signs preceding the
terms in the righthand members of equations (98) could be positive or negative. But it can be shown that by a suitable
all of change of the parameters and of the functions F and these cases reduce to (98). Thus, for example, we consider the surface defined by the equations which result when the second
3&gt;
terms of the righthand members of (98) are replaced by
In order that the surface thus defined can be brought into coin cidence, by a translation, with the surface (98), we must have
Dividing these equations, member by member, we have
from which
it
follows that
Substituting this value in the last of the above equations,
we
find
*
Zeitschrift fur
Mathematik und Physik, Vol. IX
(1864), p. 107.
MINIMAL CUEVES
257
and this value satisfies the other equations. Similar results fol low when another choice of signs is made. The reason for the
particular choice made in (98) will be seen reality of the surfaces.
when we
discuss the
Incidentally
we have proved
the theorem
:
When a minimal
and
is
surface is defined by equations (98), the necessary condition that the two generating curves be congruent sufficient
that
(99)
, )._1
(
From
(98)
we
obtain
so that the linear element
is
(100)
ds
2
= (l + uv}*F(u)3&gt;(v)dudv.
normal
We
find for the expressions of the directioncosines of the
1 H uv
1
+ uv
is
1
+ uv
and the linear element of the sphere
,
2
Alt, Also we have
(102)
4 dudv (1
d&lt;r
+
)
&gt;
D=
and of the asymp
so that the equations of the lines of curvature totic lines are respectively
(103) (104)
2F(u) du
3&gt;
(v)
F(u) du
2
+
&lt;$&gt;
= 0, dv* = 0. (v)
dv*
:
These equations are of such a form that we have the theorem
When a minimal surface is referred to its minimal lines, the finite equations of the lines of curvature and asymptotic lines are given by quadratures, which are the same in both cases.
In order that a surface be real
be
real.
Consequently u and
v
its spherical representation must must be conjugate imaginaries, as
258
is
MINIMAL SURFACES
13,
seen from (101) and
and the functions
if
F
and
&lt;
must be
denotes the real part of a conjugate imaginary. function 9, all real minimal surfaces are defined by
Hence
RO
x
= fi f (1  u
2
)
F(u) du,
y
I
=R
ft (1
+u
2
)
F(u) du,
z=R
2uF(u)du,
where F(u) is any function whatever of a complex variable u. In like manner the equations of the lines of curvature may be
written in the form
(105)
72 /
^/F(u)du
=
const.,
11
\
iVF(u)du
= const.
whether
111. Double minimal surfaces.
It is natural to inquire
the same minimal surface can be denned in more than one way by equations of the form (98). We assume that this is possible, and
indicate
by u v v^ and F^(u^ ^V^) the corresponding parameters and functions. As the parameters u^ v l refer to the lines of length zero on the surface, each is a function of either u or v. In order to
determine the forms of the latter we make use of the fact that the of positive directions of the normal to the surface in the two forms
parametric representation may have the same or opposite senses. When they have the same sense, the expressions (101) and similar ones in u v and v l must be equal respectively. In this case
(106)
%!=!*,
v^v.
the resulting equations
If the senses are opposite, the respective expressions are equal to
within algebraic signs.
(107)
From
we
find
u
compare equations
(98)
When we
u l and
vv
we
find that for the case (106)
with analogous equations in we must have
and for the case (107)
DOUBLE MINIMAL SUBFACES
Hence we have the theorem
mined by
(108)
:
259
A necessary and sufficient condition that two minimal surfaces, deter the pairs of functions F, and Fv v be congruent is that
&lt;& &lt;&
^w
on one surface corresponds the point
(
to the point (u, v)
 u
on
the other,
sense.
and
the
normals at these points are parallel but of different
In general, the functions
same.
If
members
by
and l as given by (108) are not the so also are and r Ih this case the righthand they are, of equations (98) are unaltered when u and v are replaced
&lt;
F
F
4&gt;
l/v and
1/u respectively.
v)
Hence the Cartesian coordinates
differ at
j
of the points (u,
and
(
 &gt;
most by constants.
And
so the regions of the surface about these points either coincide or can be brought into coincidence by a translation. In the latter case
periodic and consequently transcendental. Suppose that it is not periodic, and consider a point ZjJ(w V Q ). As u varies continuously from U Q to v varies from v to l/w and the point describes a closed curve on the surface by returning
the surface
is
,
l/t&gt;
,
,
to
P
Q
.
face.
But now the positive normal is on the other side of the sur Hence these surfaces have the property that a point can pass
continuously from one side to the other without going through the surface. On this account they were called double minimal surfaces
by Lie,* who was the
first to
From
the third theorem of
study them. 110 it follows that double minimal
in both systems are congruent.
surfaces are characterized by the property that the minimal curves The equations of such a surface
may
be written
The
surface is consequently the locus of the midpoints of the chords of the curve
f
=/,(),
i =/,(),
is
?=/,(),
which
lies
upon the surface and
* Math. Annalen, Vol.
the envelope of the parametric
(1878), pp. 345350.
curves.
XIV
260
MINIMAL SURFACES
EXAMPLES
1. The focal sheets of a minimal surface are applicable to one another and to the surface of revolution of the evolute of the catenary about the axis of the latter.
2.
Show
that there are no minimal surfaces with the minimal lines in one
family straight.
3. If two minimal surfaces correspond with parallelism of tangent planes, the minimal curves on the two surfaces correspond.
4. If two minimal surfaces correspond with parallelism of tangent planes, and the joins of corresponding points be divided in the same ratio, the locus of the
points of division
5.
is
a minimal surface.
that the right helicoid is defined by F(u) constant, and that it is a double surface.
6.
Show
= im/2 w2
,
where
m
is
a real
The surface
for which F(u)
=
2
is
called the surface of Scherk.
Find
its
that it is doubly periodic and that equation in the Monge form z it is a surface of translation with real generators which are in perpendicular planes.
7.
= f(x, y). Show
By definition
is
a meridian curve on a surface
tion
a great
circle
is one whose spherical representa on the unit sphere. Show that the surface of Scherk possesses
two families
of plane meridian curves.
* remarked that 112. Algebraic minimal surfaces. Weierstrass formulas (98) can be put in a form free of all quadratures. This
is
done by replacing F(u) and
accents
indicate
&lt;J&gt;(v)
the
differentiation,
where and # by f and then integrating by
"(u)
"(v),
parts.
This gives
x
p.
2
+ uf (u) f(u) +
"(u)
^i
4&gt;"(v)
+ v$(v)
h iv(
(109)
iuf(u)
+ if(u)

^
&lt;l&gt;"(v)
=
uf"(u)
) +
v&lt;f&gt;"(v)
4&gt;
(v).
It is clear that the surface so
denned
is
real
when
/
and
&lt;f&gt;
are
conjugate imaginary functions.
In this case the above formulas
may
be written
:
(110)
= R[(ly = Ri [(1 +
u*)f"(u)
u*)f"(u)
+ 2 uf (u)  2 uf (u) +
Akademie
(1866), p. 619.
* Monatsberichte der Berliner
ALGEBRAIC MINIMAL SURFACES
261
* However, it is not necessary, as Darboux has pointed out, that f and be conjugate imaginaries in order that the surface be real.
(f&gt;
For, equations (109) are unaltered
2
t
if
/and
be replaced by
f (u) =f(u) + A (1  u + Bi (1 + u*) + 2 Cu,  A(l  v + Bi ({.{ v ) 2 Cv, ^(v) =
)
2
2
(f&gt;(v)
)
where A, B, C are any constants whatever. Evidently, if / and are conjugate imaginaries, the same is not true in general of /,_
&lt;/&gt;
and
&lt;f&gt;
l
;
but the surface was real for the former and consequently
for the
latter
also.
&lt;/&gt;
It is readily found that /t and x functions only in case J, J5, C are pure are conjugate imaginary
is
real
imaginaries.
surfaces.
&lt;
Formulas (109) are of particular value in the study of algebraic Thus, it is evident that the surface is algebraic when/ and are algebraic. Conversely, every algebraic minimal surface In proving this we is determined by algebraic functions / and
&lt;/&gt;.
follow the method suggested by Weierstrass.f establish first the following lemma
We
4&gt;
:
Gttven a function $*(?
of
,
;
if in a certain
77,
4&gt;
and let domain an algebraic
*??)
j~
+
"^(f,
denote the real part relation exists between M*,
77)
and
is
an algebraic function of
+
irj.
If the point f
=
0,
rj
=
does not
lie
within the domain under
can be effected by a change of variables without the argument. Assuming that this has been done, we vitiating in a power series, thus develop the function
consideration, this
&lt;E&gt;
:
4&gt;
= a + #o + K + ibj ( +
s
irj)
+ (a + ib
2
2)
(f
+
2
irj)
+
is
.
.
.
,
where the a
and
5 s are real constants.
Evidently M*
given by
1
(a,
 ^) (f  ^) + J (a,  i6
8)
(f
ii?)
2
+


.
Let J^(^, f ?;) = denote a rational integral relation between When M* has been replaced by the above value, and the f, and 77. resulting expression is arranged in powers of  and 77, the coeffi
,
"SP,
cient of every term is identically zero. zero when f and 77 have been replaced
* Vol.
I, p.
They
will continue to be
by two complex quantities
293.
f
Monatsberichte
tier
Berliner Akademie (1867), pp. 511518.
262
a and
/3,
MINIMAL SURFACES
provided that the development remains convergent. The condition for the latter is that the moduli of a and /3 be each one
half the
modulus
of f
+
irj.
This condition
is
satisfied if
we
take
Now we
have

.

K
i5
)
+
&lt;
(f
t
irt,
f
+
ti,]
=
0,
which proves the lemma.
In applying this
(101) that
lemma
to real
minimal surfaces we note from
lZ ~Y~
to u l
X = +v _u
where u
lZ
Y= _u
2i
v,
consequently the lefthand members of these equations are equal
and
v l respectively,
= u^ + iv r When
the surface
is
algebraic there exists an algebraic relation between the functions
X
.L
Y
7 j
,
A.
/j
7
and each
of the Cartesian coordinates.*
Since, then,
an algebraic relation between u^ v^ and each of the coordinates given by (110), it follows from the lemma that each
there
is
of the three expressions
&lt;k(
= (1  ?/)/ = i (1 + u fa(u)
M)
2
+
2
uf
(u)
 2/(w),
)f"(u)
 2 iuf (u) + 2 if(u),
also isf(u)
for,
4&gt;
9 (u)
= 2uf"(u)2f(u)
and so
;
are algebraic functions of w,
Hence we have demonstrated
The necessary and
algebraic surface
* For.
if
the theorem of Weierstrass
:
sufficient condition that
equation (110) define an
is
that f(u) be algebraic.
defined by F(x. y,
z)
the surface
is
z.
=
0,
the directioncosines of the normal
are functions of x, y,
Eliminating two
of the latter
between
X
_
&gt;
Y
F(x,
y, z)
=
_
_&gt;
and
0,
we have a
relation of the kind described.
ASSOCIATE SURFACES
113.
263
Associate surfaces.
surface
S
is
are
When the equations of a minimal written in the abbreviated form (95), the linear
element
This
is
the linear element also of a surface defined by
any constant. There are an infinity of such surfaces, minimal surfaces. It is readily found that the direc tioncosines of the normal to any one have the values (101). Hence any two associate minimal surfaces defined by (111) have their tan
is
where a
called associate
gent planes at corresponding points parallel, and are applicable. Of particular interest is the surface Sl for which a = ?r/2. Its
equations are
i)du

i
C
I
(1
v
2
&lt;
)
(v)
dv,
(112)
{y =  ^ J/ &
\
(1
+u
2
)
F(u) du
i
I
*
dv.
=i
I
uF(u) du
I
v&lt;&
(v)
In order to show that
S
l
is
the adjoint
(
109) of
S,
we have only
to prove that the asymptotic lines on either surface correspond to the lines of curvature on the other. For S the l equations of the
lines of curvature
and asymptotic
lines are
2 iF(u) du
i
(v)
dv 2 =
0,
respectively.
Comparing these with (103) and
is satisfied.
(104),
we
see that
the desired condition
From
(98)
and (112) we obtain the
identities
\
dx dx
l
+
dy dy
l
+ dz dz^ =
0.
The
latter has the
following interpretation
:
On
two adjoint minimal surfaces at points corresponding with
par
allelism of tangent planes the tangents to corresponding curves are
perpendicular.
264
MINIMAL SURFACES
(105)
it
From
follows that
if
we put
/
u
the curves
curvature.
+
iv
=
^/F(u) du,
u
= const,
and
v
= const,
on the surface are
its lines
of
Moreover, for an associate surface the lines of curva
ia
ture are given by
ttf
R [e
or
2
(u
+ iv)] = const.,
.
R [ie
u
.
2
(u h iv)]
f
=
const. t
u cos
22
a
v sin
to the
a
= const.,
a
sin
22
_ v cos
#,
a 
= const.
From
The
this result follows the
lines of curvature
theorem
:
S correspond
curves on
on a minimal surface associate to a surface S which cut its lines of curvature under
the constant angle a/ 2.
Since equations (111)
may
be written
xa = x cos a

+

x l sin
(114)
.
^^ycosa + ^sina,
za z cos
a
z l sin #,
the plane determined by the origin of coordinates, a point on a minimal surface and the corresponding point on its adjoint, con
P
on every associate minimal the locus of these points a is an ellipse with its Moreover, center at the origin. Combining this result and the first one of
tains the point
P a
corresponding to
P
surface.
P
this section,
we have
*
minimal surface admits of a continuous deformation into a series of minimal surfaces, and each point of the surface describes an ellipse whose plane passes through a fixed point which is the center of the
ellipse.
A
114.
Formulas of Schwarz. Since the tangent planes
its
to a
minimal
surface and
adjoint at corresponding points are parallel,
we have
From
this
and the second of (113) we obtain the proportion
dx l
Zdy
Ydz
__ =
dy
l
Xdz
dz _ Z dx~ Y dx X dy
l
FORMULAS OF SCHWAKZ
265
In consequence of the first of (113) the sums of the squares of the numerators and of the denominators are equal. And so the com
1. If the expressions for the various quanti ratio is 1 or be substituted from (98), (101), and (112), it is found that the 1. Hence we have value is
mon
ties
(115)
dx^Ydz
Zdy, dy l
= Zdx
Xdz, dz
1
= Xdy
Ydx.
From
these equations and the formulas (95), (112)
we have
1
=x+i
*
Zdy  Ydz,
(116)
i=y +
l
(xdz \
Zdx,
= z^i = xi
Ydx
Xdy,
and
1
\
Zdy
Ydz,
(117)
^z
importance
i
\Ydx
\
X dy.
These equations are known as the formulas of Schwarz* Their is due to their ready applicability to the solution of
:
the problem
To determine a minimal surface passing through a given curve
and admitting
at each point of the curve a given tangent plane.\
In solving this problem we let C be a curve whose coordinates #, y, z are analytic functions of a parameter f, and let JT, Y, Z be analytic functions of t satisfying the conditions
X + F + Z = 1,
2 2 2
Xdx + Ydy + Zdz =
0.
*
t
Crelle, Vol.
LXXX
is
(1875), p. 291.
:
a special case of the more general one solved by Cauchy To deter mine an integral surface of a differential equation passing through a curve and admitting at each point of the curve a given tangent plane. For minimal surfaces the equation is (84). Cauchy showed that such a surface exists in general, and that it is unique unless the curve is a characteristic for the equation. His researches are inserted in Vols. XIV, XV of the Comptes Rendus. The reader may consult also Kowalewski, Theorie der partiellen Differentialgleichungen, Crelle, Vol. LXXX (1875), p. 1; and Goursat, Cours d Analyse Mathematique, Vol. II, pp. 563567 (Paris, 1905).
This problem
266
If
MINIMAL SUKFACES
,
x uJ y u
zu
denote the values of
x, y,
z
when
t
is
replaced by a
t
complex variable u, by v, the equations
and xv y v
,
,
zv
the values
when
is
replaced
(118)
l 
f"(YdxXdy) Jv
define a minimal surface which passes through C and admits at each point for tangent plane the plane through the point with
directioncosines X,
I
7
,
Z.
For,
these equations define C.
And
when u and v are replaced by the conditions (96) and
,
are satisfied.
Furthermore, the surface defined by (118) affords the unique solution, as is seen from (116) and (117).
are real, the equations of the real minimal surface, satisfying the conditions of the problem, may be ,i put in the form
When,
in particular,
C and
t
x = R\x + il (ZdyYdz}\,
y
z
= R \y + i C\Xdz  Zdx)] = R \z + i r\Ydx  Xdy\\
,
As an
straight line.
a application of these formulas, we consider minimal surfaces containing denote the angle which If we take the latter for the zaxis, and let
the normal to the surface at a point of the line
makes with the
Y=sin&lt;t&gt;,
xaxis,
we have
x
=
y
=
0,
z=t,
JT=cos0,
Z=
0.
Hence the equations
x
of the surface are
=  RiTsm
&lt;f&gt;dt,
y
=
B{J**C08^ctt,
z
= R(u).
surface.
an analytic function of t, whose form determines the character of the For two points corresponding to conjugate values of M, the zcoordinates are equal, and the x and ^coordinates differ in sign. Hence
Here
is
&lt;#&gt;
:
Every straight
line
upon a minimal surface
is
an axis of symmetry.
FORMULAS OF SCHWAKZ
EXAMPLES
1.
267
The tangents
If
to corresponding curves
on two associate minimal surfaces meet
under constant angle.
2.
corresponding directions on two applicable surfaces meet under constant
angle, the latter are associate
3.
minimal surfaces.
that the catenoid and the right helicoid are adjoint surfaces and deter mine the function F(u) which defines the former.
Show
Let
4.
C
surface (a) the equations of the
be a geodesic on a minimal surface S. Show that may be put in the form
y
=
and
X,
/*,
where
f,
77,
f are the coordinates of a point on C,
;
v
the directioncosines
of its binomial
(6)
if
denotes the curve on the adjoint S t corresponding to C, the radii of and second curvature of C are the radii of second and first curvature of C is a plane curve, the surface is symmetric with respect to its plane. (c) if C
C
first
;
5.
The surface
for
which F(u)
=
1
is
u4
called the surface of Henneberg ;
it is
a double algebraic surface of the fifteenth order and fifth class.
GENERAL EXAMPLES
1. The edge of regression of the developable surface circumscribed to two confocal quadrics has for projections on the three principal planes the evolutes of the focal conies.
2.
By definition a tetrahedral surface is one whose x = A (u  a) m (v  a), y = B(u b) m (v  6) n
equations are of the form
z
,
= C(u
c)
m
(v

n
c)
,
where A, B, 0, w, n jugate, and that the asymptotic
m = n,
the equation of the surface
III
is
are any constants. Show that the parametric curves are con lines can be found by quadratures also that when
;

c)
+

a)
+

b)
=
(a

b) (b
2,
c) (a

c).
^)&gt;
()"(c
(0"&lt;a
3.
Determine the tetrahedral surfaces, defined as in Ex.
upon which the
parametric curves are the lines of curvature.
4.
on an
5.
Find the surfaces normal to the tangents to a family of umbilical geodesies surface. elliptic paraboloid, and find the complementary
At every point
of a geodesic circle with center at an umbilical point
on the
ellipsoid (10)
abc
= fW
&lt;i
(a
+
c
_
r
^
where
r
is
the radius vector of the point
(cf.
102).
is
6. The tangent plane to the directorcone of a ruled surface along a generator distant point on the parallel to the tangent plane to the surface at the infinitely
corresponding generator.
268
7.
MINIMAL SURFACES
Upon
The
the hyperboloid of one sheet, and likewise
upon the hyperbolic parab
oloid, the two lines of striction coincide.
line of striction of a ruled surface is an orthogonal trajectory of the only in case the latter are the binormals of a curve or the surface is a generators
8.
right conoid.
9. Determine for a geodesic on a developable surface the relation existing between the curvature, torsion, and angle of inclination of the geodesic with the
generators.
Z2
and a the angle between two lines li and about the former with a helicoidal motion of parameter a surface if a = h cot a. If a = h tan a, the 62), the locus of 1 2 is a developable (cf of the binormals of a circular helix. surface is the locus
10. If h denotes the shortest distance
latter revolves
,
and the
.
11. If the lines of curvature in one family upon a ruled surface are such that the segments of the generators between two curves of the family are of the same is constant and the line of striction is a line length, the parameter of distribution
of curvature.
12. If two ruled surfaces meet one another in a generator, they are tangent to one another at two points of the generator or at every point in the latter case the central point for the common generator is the same, and the parameter of distribu tion has the same value.
;
13. If tangents be drawn to a ruled surface at points of the line of striction in directions perpendicular to the generators, these tangents form the conju line of striction as the given surface. More gate ruled surface. It has the same the normal to the surface at the central over, a generator of the given surface, and the generator of the conjugate surface through C point C of this generator, are parallel to the tangent, principal normal, and binormal of a twisted curve.
and
14.
Let
to
C
normals
S along
be a curve on a surface S, and S the ruled surface formed by the C. Derive the following results
:
distance between nearby generators of S is of the first order unless C is (a) the a line of curvature denotes the distance from the central point of a generator to the point of (6) if r
;
intersection with S,
rS (dX) 2
Z dxd
X
;
is conjugate to the tangent to the surface at to C at a point (c) the tangent line of shortest distance parallel to the and minimum values of r are the principal radii of S, pi, and (d) the maximum 2 is the where be written r = pisin 2 f p 2 cos and the above
;
M
M
p2 ,
equation
may
&lt;f&gt;
&lt;/&gt;
tf&gt;,
angle which the corresponding
line of shortest distance
pz&gt;
makes with
the tangent to
the line of curvature corresponding to
15. If
C and
(cf.
6"
are two orthogonal curves on a surface, then at the point of
intersection
Ex. 14)
1111
4
&gt;.*
rB
16. If
~tf
\
+
*
C and C
(cf.
are two conjugate curves on a surface, then at the point of
14)
j j
i
intersection
Ex.
r
R
GEKEBAL EXAMPLES
269
17. If two surfaces are applicable, and the radii of first and second curvature of every geodesic on one surface are equal to the radii of second and first curvature of the corresponding geodesic on the other, the surfaces are minimal.
1 8.
The surface
for
face of Enneper ;
(a) it is
it
in (98) is constant, say 3, which possesses the following properties
:
F
is
called the
minimal sur
unaltered
an algebraic surface of the ninth degree whose equation
;
;
is
when
x, y, z are
(6) it
(c) if
z respectively replaced by y, x, meets the plane z = in two orthogonal straight lines we put u = a i/3, the equations of the surface are
x
= 3a +
3 ap?
 a3
,
,
y
=
3
ft
+ 3 a2
ft
ft
3
,
z
= 3 a2 ;
3
2
/3
,
and the curves a
const.
ft
=
const, are the lines of curvature
(d) the lines of curvature are rectifiable unicursal curves of the third
order and
they are plane curves, the equations of the planes being
x
(e)
+
az
3a2a = 0,
3
y

ftz

3ft

2
^ = 0;
of circles
to
by a double family whose planes form two pencils with perpendicular axes which are tangent the sphere at the same point
;
the lines of curvature are represented on the unit sphere
(/) the asymptotic lines are twisted cubics (g) the sections of the surface by the planes
;
are double curves on the surface
x= and y = are cubics, which and the locus of the double points of the lines of
curvature
;
(h) the associate
through the angle
(i)
minimal surfaces are positions of the original surface rotated 113 a/2, about the zaxis, where a has the same meaning as in
;
the envelope of the plane normal, at the midpoint, to the join of any two points, one on each of the focal parabolas
is
the surface
X
=
4
cr,
y
=
0,
z

2
a2 
1
;
x

0,
y
=
4
ft,
z
= 12 ft2

the planes normal to the two parabolas at the extremities of the join are the planes of the lines of curvature through the point of contact of the first plane.
19. Find the equations of Schwarz of a minimal surface when the given curve an asymptotic line.
20.
is
Let S and S be two surfaces, and
;
let the points at
which the normals are
parallel correspond
for convenience let
S and S be
;
referred to their
common con
jugate system. Show that if the correspondence is conformal, either S and S are homothetic or both are minimal surfaces or the parametric curves are the lines of curvature on both surfaces, and form an isothermal system.
;
Find the coordinates of the surface which corresponds to the ellipsoid after 20. Show that the surface is periodic, and investigate the points corresponding to the umbilical points on the ellipsoid.
21.
the
manner of Ex.
the equations of an ellipsoid are in the form (11), the curves u + v = on spheres whose centers coincide with the origin and at all points of such a curve the product pW is constant ( 102).
22.
When
const, lie
;
CHAPTER
VIII
SURFACES OF CONSTANT TOTAL CURVATURE. WSURFACES. SURFACES WITH PLANE OR SPHERICAL LINES OF CURVATURE
115. Spherical surfaces of revolution.
Surfaces whose total cur
vature
K
(
is
the same at
all
points are called surfaces of constant
surfaces of this kind are called
curvature.
When
this constant value is zero, the surface is devel
opable
64).
The nondevelopable
is positive or negative. spherical or pseudospherical, according as consider these two kinds and begin our study of them with
K
We
the determination of surfaces of revolution of constant curvature.
When
upon a surface
of revolution the curves v
= const,
are
is
the meridians and u
= const,
the parallels, the linear element
reducible to the form
(1)
d8*=du*+Gdif,
where
G
is
a function of
u alone
(
46).
is
In this case the expres
sion for the total curvature (V, 12)
(2)
K=
,
2 For spherical surfaces we have 7f=l/a where a is a real constant. Substituting this value in equation (2) and integrating, we have
(3)
constants of integration. From (1) it is seen that a change in b means simply a different choice of the parallel u = 0. If we take 6 0, the linear element is
where
b
and
c are
(4)
ds
2
=du + c
2
2
cos
2
^ a
.
2
.
From
(5)
(III, 99,
100)
r=
it
follows that the equations of the meridian
curve are
u
cos&gt;
z
a
it = C \1 J\
/
c
2
.
a2
jsma
9
u
270
SPHERICAL SURFACES OF REVOLUTION
and that
v
271
measures the angle between the meridian planes. There are three cases to be considered, according as c is equal
greater than, or less than,
a.
to,
CASE
r
I.
c
= a. Now
z
= a cos)
a
u a sin a
.
&gt;
and consequently the surface is a sphere.
CASE
for z
it
II. c
&gt;
a.
From
2
the expression
follows that sin
&gt;
a
&lt;
1
and con
0. Hence the surface is sequently r made up of zones bounded by minimum
parallels
whose
radii are equal to the
?/
FIG. 26
minimum
value of cos ;
and the greatest
parallel of each zone
is
of radius c
as in
&lt;
fig.
26,
where the curves represent geodesies.
from
to
c,
CASE
III. c
a.
is any odd integer. At these mcnr/2, where ing to the value u on the axis the meridians meet the latter under the angle points
v?
Now r varies =
is
the former correspond
m
1
sin"
. a
27).
Hence
the surface
made up
of a series of spindles
z
(fig.
For the cases II and III the expression for
can be
integrated in terms of elliptic functions.* It is readily found that these two surfaces are
applicable to the sphere with the meridians and parallels of each in correspondence. Thus, if we
write the linear element of the sphere in the form
ds
it
2 2 du 2 4 a 2 cos 2  dv
,
a
follows from (4) that the equations
u
FIG. 27
= u.
It is evident that for values of b other
determine the correspondence desired. than zero we should be
results.
brought to the same
However,
I,
for the sake of future
*Cf. Bianchi, Vol.
p. 233.
272
reference
SURFACES OF CONSTANT CURVATURE
we
write
down
when
b
=
7r/2
and
(i)
the expressions for the linear element Tr/4 together with (4), thus
:
ds
2
=du*
(6)
(ii)
ds =du*2
2
(iii)
ds
=du*
cos
u TT\ a
,
}dv\ 4/
Let S be a surface with the linear element (6, i), and consider the zone between the parallels u = const, and rt 1 = const. A point
of the zone
is
determined by values of u and v such that
The parametric
are such that
values of the corresponding point on the sphere
9
_
the given zone on S does not cover the zone on the sphere between the parallels M O = const, and u^ = const. a it not only covers it, but there is an overlapping. but when c
Hence when
c
&lt;
,
;
&gt;
pseudospherical
116. Pseudospherical surfaces of revolution. In order to find the 2 in (2) by surfaces of revolution we replace I/a
K
and
integrate.
This gives
V5 =
where
ct
c.
cosh a
+
&lt;?_
sinh 
&gt;
a
and
c 2 are
constants of integration.
We
consider
first
the particular forms of the linear element arising when either of these constants is zero or both are equal. They may be written
ds
2
(i)
= =
du
?/
a
(ii)
+c
oU 2 sinh i
a
(iii)
ds^dtf+fe" dv*.
Any case
other than these
may be
obtained by taking for
either
of the values cosh

or sinh(
where
b is a constant.
PSEUDOSPHEKICAL SURFACES OF REVOLUTION
By
273
a change of the parameter u the corresponding linear elements are reducible to (i) or (ii). Hence the forms (7) are the most general.
The corresponding meridian curves
are defined
C
by
= c cosh r
&gt;
2
=
\
sum2  aw
.
,
U
,
;
(8)
(ii) v
w = tfsinha
2= C \ 1 JN
I
&lt;?
a
2
.u costfdu;
a
(iii)
r
= cea
.
z
=
We
and
consider these three cases in detail.
I.
CASE
0.
c.
The maximum and minimum values
of sinh
2
 are a 2 /e 2
a
Hence the maximum and minimum values
of r are
Va + c
2
2
and
At
points of a
maximum
parallel the tangents to the
merid
ians are perpendicular to the axis, and at points of a minimum parallel they are par allel to the axis. Hence the former is a cus
pidal edge, and the latter a circle of gorge, so that the surface is made up of spoollike
sections.
It is represented by fig. 28, upon which the closed curves are geodesic circles and the other curves are geodesies. These
pseudospherical surfaces are said to be of the hyperbolic type.*
CASE
c
2
II.
In order that the surface be real
,
2 cannot be greater than a a restriction not necessary in either of the other cases.
we put e = asino:,f the maximum and minimum values of cosh2 are cosec 2 o; and 1, and
If
FIG. 28
a
the correspond
ing values of r are a cos a and 0. The tangents to the meridians at points of the former circle are perpendicular to the axis, and at the points for which r is zero they meet the axis under the angle a.
Hence the surface
.
is
made up
I, p.
of a series of parts similar in shape
f Cf.
* Cf Bianchi, Vol.
223.
Bianchi, Vol.
I,
p. 220.
274
SURFACES OF CONSTANT CURVATURE
is
to hourglasses.
the curves
Fig. 29 represents one half of such a part one of an asymptotic line and the others are parallel geodesies.
;
The
surface
is
called a pseudospherical surface of the elliptic type.
CASE
III.
equations of the
In the preceding cases the meridian curve can
be expressed without the quadrature sign by means of elliptic functions.*
In this case the same can be done by
means
if
of trigonometric functions.
sin
d)
For,
we put
=
a
ea.
equations
FIG. 29
(9)
(iii)
of (8)
2
become
cos(/&gt;).
r
=
asin&lt;,
= a (log tan^f
We
find that
point makes curve. Since the length of the segment of a tangent between the point of contact and the intersection with the axis is r cosec
or
a,
the angle which the tangent to a meridian at a with the axis. Hence the axis is an asymptote to the
is
c/&gt;
c/&gt;
the length of the segment is independent of the point of contact. Therefore the meridian curve is a tractrix. The surface
is
of revolution of a tractrix about its asymptote sphere, or the pseudospherical surface of the parabolic type. The surface is shown in
fig.
called the pseudo
30,
which
also
pictures
a family of
line.
parallel
geodesies and an asymptotic
If the integral (3) be written in the
form
=
the cases
(i),
c,
cos
u a
f
c sin
a
1
of (6) are seen to correspond to the similar cases of (7). shall find other marks of similarity between
(ii),
(iii)
We
these cases, but
now we
desire to call at
FIG. 30
tention to differences.
of the three forms (7) determines a particular kind of in value pseudospherical surface of revolution, and c is restricted
Each
*Cf. Bianchi, Vol.
I,
pp. 226228.
APPLICABILITY
only for the second case.
275
On
the contrary each of the three forms
(6) serves to define any of the three types of spherical surfaces of revolution according to the magnitude of c.
(IV, 51) we find that the geodesic curvature of the par allels on the surfaces with the linear elements (7) is measured by
From
the expressions
.
1 a
,  tann
.
*
i
a
1 .,  cotn M a a
,
1 
,
a
Since no two of these expressions can be transformed into the other if u be replaced by u plus any constant, it follows that two
pseudospherical surfaces of revolution of different types are not applicable to one another with meridians in correspondence.
show
that
of
Applicability. Now we shall that in corresponding cases of (6) and (7) the parametric geodesic systems are of the same kind, and then we shall prove
117. Geodesic parametric systems.
when such a geodesic system is chosen for any surface constant curvature, not necessarily one of revolution, the linear element can be brought to the corresponding form of (6)
or
(T).
place we recall that when on any surface the curves const, their orthogonal trajectories, are geodesies, and u the linear element is reducible to the form (1), where G is, in
In the
first
v
= const,
=
general, a function of both u and v ; and the geodesic curvature of the curves u const, is given by (IV, 51), namely
p
ff
When,
is
in particular, the curvature of the surface is constant, 2 given by equation (2) in which may by replaced by l/a
K
.
Hence,
(11)
for spherical surfaces, the general
form of
V&
is
a and for pseudospherical surfaces
&lt;/&gt;
V& =
VG =
(v)
cos

+
A/T
(v)
sin
a
,
(12)
&lt;
(v)
cosh ci
+
i/r
(v)
sinh
a
,
where
&lt;
and
i/r
are, at
most, functions of
(7).
v.
We
consider
now
the
three cases of (6) and
276
SURFACES OF CONSTANT CUKVATUKE
I.
CASE
From
cv.
the forms
(i)
of (6)
and
(7),
and from
its
(10), it
follows that the curve u
=
is
a geodesic and that
arc
is
Moreover, a necessary and sufficient condition u = on any surface with the linear element (1) that the curve
satisfy these conditions is
measured by
=o.
Applying these conditions to (11) and forms (i) of (6) and (7) respectively.
(12),
we
are brought to the
CASE
II.
The forms
(ii)
of (6)
and
(7) satisfy
the conditions
= 0,
which are necessary and geodesic polar, in which
system be measures angles (cf. VI, 54). When these conditions are applied to (11) and (12), we obtain (ii) of (6)
sufficient that the parametric
cv
and
of (7) respectively.
III.
For (iii) of (6) the curve u = has constant geodesic curvature I/a, and for (iii) of (7) all of the curves u = const, have the same geodesic curvature I/a. Conversely, we find from is satisfied on any sur (11) and (12) that when this condition
CASE
face of constant curvature the
linear
element
is
reducible
to
one of the forms
the theorem
:
(iii).
We
gather these results together into
The linear element of any surface of constant curvature
to the
is
reducible
forms
(i),
(ii),
(iii)
of (6) or (7) according as the parametric
a point, or are geodesies are orthogonal to a geodesic, pass through to a curve of constant geodesic curvature. orthogonal
the linear element of a surface of constant curvature is in one of the forms (i), (ii), (iii) of (6) and (7), it is said to be of
the hyperbolic,
elliptic,
When
The above theorem may be
or parabolic type accordingly. stated as follows
:
is applicable to a sphere spherical surface of curvature l/a that to a family of great circles with of radius a in such a way
z
Any
the
same diameter there correspond
the geodesies orthogonal to a
APPLICABILITY
given geodesic
277
point of curvature I/ a.
it,
on the surface, or all the geodesios through any or those which are orthogonal to a curve of geodesic
surface of curvature
Any pseudo spherical
I/a
pseudospherical surface of revolution of any
according as the latter surface
is
applicable to a of the three types ;
is
2
of the hyperbolic,
elliptic,
or
par
abolic type, to its meridians correspond on the given surface geodesies
which are orthogonal
orthogonal
to
a geodesic, or pass through a point, or are a curve of geodesic curvature I/a.
to
In the case of spherical surfaces one system of geodesies can
satisfy all three conditions
circles
for in the case of the sphere the great with the same diameter are orthogonal to the equator, pass through both poles, and are orthogonal to two small circles of
;
radius a/V2, whose geodesic curvature is I/a. But on a pseudospherical surface a geodesic system can satisfy only one of these conditions. Otherwise it would be possible to apply two surfaces
of revolution of different types in
parallels correspond.
such a way that meridians and
From the foregoing theorems it follows that, in order to carry out the applicability of a surface of constant curvature upon any one of the surfaces of revolution, it is only necessary to find the
geodesies on the given surface.
set forth in the
The nature
of this problem
is
theorem
:
The determination of the geodesic lines on a surface of constant curvature requires the solution of a Riccati equation.
In proving this theorem we consider
first
defined in terms of any parametric system. It a sphere of the same curvature with center
a spherical surface is applicable to
at
the
origin.
The
tion
lines
coordinates of
u, v,
this
sphere,
expressed as functions of the
parameters
(
can be found by the solution of a Riccati equa
great circles on the sphere correspond geodesic hence the finite equation of
;
65).
To
is
on the spherical surface
ax
the
geodesies constants.
+
by
is
+
cz
=
0,
where
a,
b,
c
are arbitrary
When
the
surface
pseudospherical
we
use an imaginary
is
sphere of the
same curvature, and the analysis
similar.
278
SURFACES OF CONSTANT CURVATURE
Let a spherical surface of
2
118. Transformation of Hazzidakis.
curvature I/a
ters.
be defined in terms of isothermalconjugate
parame
Then
*
D
D"
1
and the Codazzi equations (V, 13
)
reduce to
1
dE
T
dG
_
^dF
/


II
dv
dv
du
:
From
The
these equations follows the theorem
lines of curvature of a spherical surface
form an
isothermal
conjugate system.
For, a solution of these equations
is
E G = const.,
When
(15)
F0.
G=
cosh
2 a sinh 2
a&gt;.
this constant is zero the surface is a sphere because of (13). this case,
Excluding
we
2
replace the above by
a&gt;,
E= a
2
cosh
F=
0,
Now
(16)
D D n = a sinh
a&gt;
a&gt;.
When
these values are substituted in the Gauss equation (V, 12),
namely
2
it is
#
I
a^
L^
^ H ^ HE
o&gt;
2
_
+
du \
du ULE jv
R
found that
must
a
ft&gt;
satisfy the equation
2
/18}
a H + smh dv du
o)
2

a)
cosh
o&gt;
A = 0.
2
the quantities (15) Conversely, for each solution of this equation and (16) determine a spherical surface. and v respec If equations (14) be differentiated with respect to u be added, we have tively, and the resulting equations
(19)
^ + 0^
""
du
2
dv
2
a change of sign gives a surface * The ambiguity of sign may be neglected, as metrical with respect to the origin.
sym
TRANSFORMATION OF HAZZIDAKIS
In consequence of (14) equation (17)
4
is
279
reducible to
H
4
(
\\du]
\dv / J
L#ti $ v
dv
du
Equations (14) are unaltered
sign of
if
E and G
is
F
be interchanged and the
be changed.
The same
:
true of (17) because of (19)
and
(20).
Hence we have
If the linear element of a spherical surface referred conjugate system of parameters be
ds
2
to
an isothermal
= E du
2
+2F dudv + G dv
2F dudv f E dv
2
,
there exists a second spherical surface of the same curvature referred to a similar parametric system with the linear element
ds 2
= Gdu
2
2
,
surface moreover, the lines of curvature correspond on the two surfaces.
and with
the
same second quadratic form as
the given
;
The latter fact is evident from the equation of the lines of curva ture (IV, 26), which reduces to Fdu 2 + dv 2 = 0. (G E) dudv From (IV, 69) it is seen that the linear elements of the spherical representation of the respective surfaces are

F
da 2 =
da 2 =
(Gdu
CL
2
2F dudv + E dv
+ ZFdudv
:
2
),
(E du
a/
2
f
G dv ).
2
In particular
we have
co
2
the theorem
Each
solution
of curvature
I/ a
;
of equation (18) determines two spherical surfaces the linear elements of the surfaces are
2 2
ds
ds
=a =a
2
2
(cosh
2
co
2
(sinh
co
du 2 + sinh 2 co dv 2 ), du 2 f cosh 2 co dv),
and of
,
their spherical representations
{
9
,
v
j
da 2 = sinh 2 co du 2 f cosh 2 co dv\ 2 cosh co du 2 + sinh 2 co dv 2
d&lt;r*=
;
moreover, their principal radii are respectively
pl
= a coth = a tanh p[
&),
co,
p2
p2
=
a tanh
a coth
co,
co.
280
SURFACES OF CONSTANT CURVATURE
Bianchi * has given the name Hazzidakis transformation to the relation between these two surfaces. It is evident that the former
theorem defines
this transformation in a
more general way.
119. Transformation of Bianchi.
spherical surface of curvature
conjugate parameters.
We
We consider now a pseudo2 I/a defined in terms of isothermalhave
,
H~
D__
#_
H~~
to
_1
a
4
and the Codazzi equations reduce
^+ du 2^=0, du
dv
^+2^=0. du
dv dv
These equations are
(22)
&&gt;
satisfied
2
by the values
E= a cos
w,
^=0,
G = a2 sin
of the
2
to,
where is a function which, because must satisfy the equation
/c
.
Gauss equation (V,
12),
n
,
(23)
___ =
o2
&lt;"2
&&gt;
(0
8in
coB.
Conversely, every solution of this equation determines a pseudospherical surface whose fundamental quantities are given by
(22) (24)
and by
D=
dois
2
D"
=
2
a sin
w
cos
to.
Moreover, the linear element of the spherical representation
(25)
is
=sin
2
o&gt;c^
+cos
2
a&gt;dv
2
.
f
There
ilar
to the
not a transformation for pseudospherical surfaces sim Hazzidakis transformation of spherical surfaces, but
there are transformations of other kinds which are of great im portance. One of these is involved in the following theorem of
Ribaucour
:
If in the tangent planes 2 I/ a circles of radius a
surfaces of curvature
* Vol.
t
a pseudo spherical surface of curvature be described with centers at the points of
to
contact, these circles are the orthogonal trajectories of
an
infinity of
1J a
made
2
.
II, p. 437.
is
This choice of sign ary form.
so that the following formulas
may have
the custom
TKANSFOKMATION OF BIANCHI
281
In proving this theorem we imagine the given surface S referred to its lines of curvature, and we associate with it the moving trihe
dral
whose axes
rt
are tangent to the parametric lines.
From
(22)
and (V, 75, 76)
it
follows that
P
P\
= cos w = a cos
ft),
= sin
t] l
,
t
=s 0,
n
r
= d(0
cv
77
,
r1
= 3(0
vU
=a
sin
&&gt;,
fx
==
0.
In the tangent z^plane we draw from the origin a segment of length #, and let 6 denote its angle of inclination with the #axis. The coordinates of the other extremity 1 with respect to these
M
M
axes are a cos 0, a sin 0, 0,
a displacement of
M
l
as
M
and the projections upon these axes of moves over S are, by (V, 51),
a
L
sin 6
dO
f
cos
oadui
\dv
du
rfw
\
du
dv } sin 6 L
I J
a cos 6 d6 f sin
L
&) c?y
+(
&)
+
\cv
cos
du
dv } cos
/
J
,
a [cos
&&gt;
sin
c?v
sin
du\.
We seek
line
now
l
the conditions which
must
l
MM be tangent to the locus of M denoted by S^ and that the tangent plane to S at M be perpendicular to the tangent plane to
l
satisfy in order that the
1
S at M. Under these conditions the directioncosines of plane to S with reference to the moving trihedral are
l
the tangent
(26)
sin0,
COS0,
0,
and since the tangent to the above displacement must be in plane, we have
(27)
this
dO
this
+(
\dv
 sin
cos
co] /
du+(+ cos \cu
all
sin
a&gt;\
dv
=
0.
)
Jtf",
As
equation must hold for
displacements of
it
is
These equations satisfy the condition of integrability in conse is a solution of equation (23), as is quence of (23). Moreover, seen by differentiating equations (28) with respect to u and v
respectively and subtracting.
282
SURFACES OF CONSTANT CURVATURE
of (28) the above expressions for the projections of a of M^ can be put in the form displacement
By means
a cos a sin
a (cos
(cos (cos
&)
ft)
o&gt;
+ sin cos 9 du + sin
cos 6
du
&&gt;
sin 6 dv),
sin 6 dv),
o&gt;
sin #
c?t&gt;
sin
&)
cos $ du).
From
these
it
follows that the linear element of
S
l
is
=a ds?
In order to prove that
its lines
2
(cos 9
is
2
du
2
+ sin
2
2
cfrr
).
S
l
of curvature, it remains for us to
a pseudospherical surface referred to show that the spherical
representation of these curves forms an orthogonal system. obtain this representation with the aid of a trihedral whose vertex
is
We
fixed,
and which rotates so that
its
axes are always parallel to
the corresponding axes of the trihedral for S. The point whose coordinates with reference to the new trihedral are given by (26) serves for the spherical representation of Sr The projections upon
these axes of a displacement of this point are reducible, by means
of (28), to
cog e ^ cos
sin #(cos sin
&)
m
&)
sin sin
du du
_ sin a cos
sin
ft)
&lt;w
dv ^
dv),
cos
sin
du
cos
cos 9 dv,
is
from which
it
follows that the linear element
Since
is
curvature
a solution of (23), the surface Sl is pseudospherical, of 2 and the lines of curvature are parametric. To 1/
,
each solution 9 of equations (28) there corresponds a surface Sr Darboux * has called this process of finding S1 the transformation
the complete integral of equations (28) involves an as remarked arbitrary constant, there are an infinity of surfaces
of Bianchi.
&gt;S\,
As
by Ribaucour.
(29)
Moreover,
if
we put
*tan.
&lt;.
these equations are of the Riccati type in
Hence, by
14,
one transform of Bianchi of a pseudospherical surface is known, the determination of the others requires only quadratures.
* Vol. Ill, p. 422.
When
TRANSFORMATION OF BIANCHI
From
$! are
(III, 24)
it
283
follows that the differential equation of the
curves to which the lines joining corresponding points on
S and
tangent
is
(30)
cos
co
smddu
sin
o&gt;
cosddv
= 0.
Hence, along such a curve, equation (27) reduces to
7/1
d6\
 du H dv
da)
,
d(o
i
f.
0.
dv
du
But from
geodesies have the values (22).
(VI, 56) it is seen that this is the Gauss equation of upon a surface whose first fundamental coefficients
Hence
:
The curves on S
to
which the lines joining corresponding points on
S and S
l
are tangent are geodesies.
trajectories of the curves (30) are defined
The orthogonal
(31)
by
coswcostfdtt
+
sinw
sinflcto
=
0.
In consequence of (28) the lefthand
member of
this equation is
an exact differential.
d
the quantity
=
a (cos w cos 0du
+
sin
w
sin 6dv),
(30).
e~& a
is
an integrating factor of the lefthand member of
rj
Conse
quently
we may
define a function
drj
thus
:
=
ae~
/a
(cos
w
sin 6 du
sin
w
cos 6 dv)
.
In terms of
(32)
and
i\
the linear element of
2
&lt;Zs
S
is
expressible in the parabolic form
cfys.
(7),
=
d
2
+ e^A
Equation (31) defines also the orthogonal trajectories of the curves on Si which the lines MMi are tangent, and the equation of the latter curves is
sin
to
w cos 6 du
cos w sin 6 dv
=
0.
The quantity
e* /a is
an integrating factor of
d
this equation,
and
dv)
,
if
we put accordingly
=
ae /a (sin w cos 6 du
cos
a;
sin
the linear element of Si
(33)
may
be expressed in the parabolic form
)
dsf
= dp + e*/ a dp.
form of the linear element of a surface of by quadratures. Hence
:
As the expressions
(32)
and
(33) are of the
revolution, the finite equations of the geodesies can be found
When
a Bianchi transformation
is
known for a
surface, the finite equation of
its
geodesies can be
found by quadratures.
117.
This follows also from the preceding theorem and the last one of
284
SURFACES OF CONSTANT CURVATURE
The transformation of Bianchi
120. Transformation of Backlund.
is
only a particular case of a transformation discovered by Backlund,* by means of which from one pseudospherical surface S another S^
of the
same curvature, can be found.
Moreover, on these two sur
faces the lines of curvature correspond, the join of corresponding and is of constant points is tangent at these points to the surfaces
meet under length, and the tangent planes at corresponding points
constant angle.
We
case,
refer
S
to the
same moving trihedral
o&gt;axis.
and the angle X and 6 denote the length of l The coordinates of 3/x are which the latter makes with the X cos 0, X sin 0, 0, and the projections of a displacement of l are
and
let
MM
as in the preceding
M
X
(34)
sin
d0
f
a cos
wdu
\
sin 6
(
\0t&gt;
du\
(?M
dv }
/
,
\cosOdO
X (cos
ft)
f
a sin&xi*;
sin
+
o&gt;
X cos#(
\dv
.
du
\
dv
du
),
I
sin 6 dv
cos 6 du)
S
denotes the constant angle between the tangent planes tP and Jft respectively, since these planes are to inter and Sl at
If
cr
M
sect in
MMv
the directioncosines of the normal to
sin
S
l
are
&
sin 0,
sin
a cos 0,
cos a.
Hence
must
satisfy the condition
a
X
sin
dB
f
a sin
or
(cos
G&gt;
sin 6 du
7
sin
&&gt;
cos
dv}
X
sin
&lt;r
du H dv
,
\dv
4
cu
&)
X cos
cr
(sin
cos
6du
cos
&)
sin 0dv)
=
0.
it is
Since this condition must be satisfied for every displacement,
equivalent to
X X
sin
a
(
\dw
/Q /I
[) = # sin
fltf/
a
cos
&)
sin 6
X cos a
sin
&)
cos
0,
sin
&lt;r
(
h
v
Q
\
)
=
a sin
&lt;r
sin
w
cos
+
X cos a cos
&)
sin 6.
cu
*Om
(1883).
of
New
Universitets Arsskrift, Vol. XIX ytor med konstant negativ krokning, Lunds Miss Emily Coddington English translation of this memoir has been made by York, and privately printed.
An
TRANSFORMATION OF BACKLUKD
If these equations
285
be differentiated with respect to v and u respect ively, and the resulting equations be subtracted, we have
a sin
2
2
crX2 =0,
a constant.
from which
erality
it
follows that
X
is
Without
loss of
gen
we
take X
= a sin
dco\
(
cr.
If this
value be substituted in the
above equations,
.
we have
f
sin
cr
(d6
\du
= sin a cos co
.
cos
a & cos 6
.
sin
o&gt;,
dv/
1
(35)
smcr
(
)
=
cos
sin
&lt;w
+ cos
er
sin
cos
co,
\dv
du/
and these equations satisfy the condition of integrability. If they be differentiated with respect to u and v respectively, and the is a solution resulting equations be subtracted, it is found that
of (23).
In consequence of (35) the expressions (34) reduce to
a cos
(cos
&)
cos
f
cos a sin
&&gt;
sin 0)
&&gt;
du
cos a cos
C?M
&&gt;
+
a cos
(cos
co
a sin
cos
cr
(sin
cos
sin 0) dv,
sin
sin
&)
cos 0)
&)
+
a sin cr(cosft) sin0o?v
&lt;*
sin #(sin
sin 6
+
cos
cr
cos
&)
cos
sin
CD
cosOdu),
and the
linear element of
^
2
is
2
d**
=a
(cos
&lt;9
dw 2 f
sin
2
In a manner similar to that of
119
it
can be shown that the
spherical representation of the parametric curves is orthogonal, and consequently these curves are the lines of curvature on S^
Equations (35) are reducible to the Riccati form by the change of variable (29). Moreover, the general solution of these equations involves two constants, namely cr and the constant of integration.
Hence we have the theorem
:
integration of a Riccati equation a double infinity of pseudospherical surfaces can be obtained from a given surface of this kind.
By the
We
it
refer to this as the transformation of Backlund,
and indicate
by
Bv
,
thus putting in evidence the constant
cr.
286
121.
SURFACES OF CONSTANT CURVATURE
Theorem
of permutability.
,
Let Sl be a transform of S by
is
means
of Sj,
of the functions (0 X
o^).
Since conversely S
a transform
and the equations
for the latter similar to (35) are reducible
to the Riccati type, all the transforms of Sl can be found by quad ratures. But even these quadratures can be dispensed with because of the following theorem of permutability
due
to
Bianehi*:
If
S and S
1
2
are transforms of
of functions (0 1?
a^ and
is
l
(0 2 ,
&lt;r
2 ),
S by means of the respective pairs a function can be found without
&lt;f&gt;
quadratures which
(&lt;,
such that by means of the pairs
2
f
&lt;r
((/&gt;,
2)
and
oj)
the surfaces
pseudospherical
S and S surface S
.
respectively are transformable into a
By
hypothesis
sin oJ
is
&lt;/&gt;
a solution of the equations
*
\H
I
sin
&lt;T
9( 2
*/)= /p ^ 4 ^ =
^/l
\
)

+
sin
$
cos
l
cos
cr
2
cos
(/&gt;
sin 0^
cos
6
sin 0.
+ cos
cr 9
sin
6
cos ^,
and
also of the equations
pi
sin
&lt;7,
p
l*
4


= sn =
d&gt;
cos
9
cos a. cos
6 sn
(37)
cos
&lt;&gt;
sn
+ cosoSn) cos
projections of the line If^Tf on the tangents to the lines of and are correspond curvature of Sl and on its normal, where l
The
M
M
1
ing points on
(38)
$ and S
l
,
are
a sin
&lt;r
2
cos
&lt;/&gt;,
a sin
&lt;r
2
sin
(#&gt;,
;
0.
of
The directioncosines of the tangents to the lines of curvature S with respect to the line JOf1? the line MQ^ perpendicular to the latter and in the tangent plane at J/, and the normal to S are
l
cos
sin
to,
cos
&lt;r
1
sin
o&gt;,
sin
sin
&lt;T
I
sin
&),
a),
cos
it
cr
1
cos w,
oj
cos w.
From
these and (38)
follows that the coordinates of
to
&lt;r
M
with
respect to
MM^ MQ^
l
and the normal
(j&gt;
S
2
are
oj
a [sin
a
+ sin
cr
2
cos
[sin
o1
a))],
a [sin
&lt;r
cos
sin
(&lt;f&gt;
&))],
sin
2
sin
(&lt;/&gt;
w)].
* Vol. II, p. 418.
THEOREM OF PEEMUTABILITY
Hence
the coordinates of
287
M
1
with respect to the axes of the moving
trihedral for
S
are
x
= cos 0j sin = sin
= sin
cr
o
l
4
cos
&lt;r
l
sin
& 2 cos
(&lt;
(&lt;
&&gt;)
sin 0, sin
2
cos o^ sin
&lt;w),
(39)
l
sin
cr
1
+ sin
&lt;7
6 l sin
cr
2
cos
((/&gt;
+ cos 0j sin
sm &
a
If
2
cos o^ sin ($
&lt;w),
sin (9
$2
the coordinates
be transformed by means of a l and the same function c, of the resulting surface can be obtained
x",
z"
y",
from
by interchanging the subscripts 1 and 2. Evidently z are equal. necessary and sufficient condition that x\ y be equal to respectively is
(39)
f
and
z
A
#",
y"
cos
1
(d
r
x")
cos
If the
[sin
2 (x
x")
+ sin B^(y + sin (y
z
y")
0,
r
=
y")
0.
above values be substituted in these equations, we obtain
((/&gt;
a l cos (# 2 0^ sin o 2 ] cos sin a l cos cr 2 sin (# 2 0^ sin
&lt;7
co)
= sin a
(^&gt;
&&gt;)
l
sin
cr
2
cos (Q n
#J,
[sin
f
2
sin
cos (^ 2 cos 2
&lt;7
0j)
cr
1
siu crj cos
2
(&lt;/&gt;
o&gt;)
sin(^
^)sin(^)
w)
= sin
cr
z
sin a^ cos(# 9
)
6^).
Solving these equations with respect to sin
and cos
(&lt;/&gt;
(&lt;/&gt;
&&gt;),
we get
sin
,.
cos
(&lt;f&gt;
(0)
sin =
sin
^
o,
oj
sin
sin
&lt;7
2
cos (^ 2
o
+ (cos
cos (0 2
/i
&lt;r.
coscr 9
^

cr
1
sin
&lt;r
2
c/j)
 ^ 2
X)
+ cos
cos
oj
cos
o
1
l)cos(^ 9
"
^.)
4
cos
r
1
cr
2
These two expressions satisfy the condition that the sum of their satisfies equations (36) and squares be unity, and the function
(j&gt;
(37).
Hence our hypotheses
is
are consistent
and the theorem
of
permutability
demonstrated.
We
may
replace the above equations by
288
SURFACES OF CONSTANT CURVATURE
result
The preceding
When
may be
expressed in the following form
:
all the transformations of the
the transforms of a given pseudospherical surface are known, former can be effected by algebraic
processes and differentiation.
Thus, suppose that the complete integral of equations (35)
(41)
is
=/(w,
is
v,
&lt;r,
c),
and that a particular integral
^i=/(
v
&gt;
*v
c i)i
of the constants, and let corresponding to particular values denote the transform of S by means of ^ and r All the trans where and formations of S are determined by the functions
&lt;7
^
&lt;r,
l
&lt;f&gt;
cr has the value r For all values Exceptional cases arise when = to + WTT, where m is an of c other than c l formula (42) gives $ odd integer. When this is substituted in equations (36) they re duce to (35). In this case S coincides with S. We consider now the remaining case where c has the value c 1? In is indeterminate. whereupon the righthand member of (42) order to handle this case we consider c in (41) to be a function of
&lt;r
o,
reducing to
cl
for
&lt;r
= a^
L
If
the function tan
for
to
Ism
~
we apply
*
the ordinary methods to
which becomes indeterminate
a
&lt;r,
=
o
numerator and denominator with respect v differentiating
we have
or
tan
/6w\ = sin ^
.
^ /a/ 4 c V
.
/

,
where
c is
an arbitrary constant.
satisfies the
*
It is necessary to verify that this
is
value of ^
equations (36), which
Cf Bianchi, Vol.
.
easily done.*
II, p. 418.
TRANSFORMATION OF LIE
122. Transformation of Lie.
289
of pseudo
Another transformation
is analytical in character was spherical surfaces which, however, Lie.* It is immediate when the surface is referred discovered by
to its
asymptotic
lines,
or to any isothermalconjugate system
of lines.
Since the parameters in terms of which the surface is defined in 119 are isothermalconjugate, the parameters of the asymptotic lines may be given by
In terms of these curvilinear coordinates the linear elements of the surface and its spherical representation have the forms
ds
da2
2
= a (da + 2 cos = da 2 cos 2
2
2
2
ft)
&)
dad/3
f
+ d/3
2
),
2
dad/3
and equation
(23) takes the
form
sin
&)
cos
ft).
dad/3
the form of this equation a solution, so also is co l =
From
it is
evident that
9
if
&&gt;
=
&lt;(#,
ft)
be
constant. ft/m) where m is any Hence from one pseudospherical surface we can obtain an infinity of others by the transformation of Lie. It should be remarked,
(f&gt;(am,
however, that only the fundamental quantities of the new surfaces are thus given, and that the determination of the coordinates re the solution of a Riccati equation which may be different
quires from that for the given surface. Lie has called attention to the fact that every Biicklund trans
formation
is
a combination of transformations of Lie and Bianchi.f
In order to prove this
equations (35)
we
effect the
change of parameters (43) upon
and obtain
d
,
n
da
(44)
d
(0 v
1 + cos a Q sin (6 + co) =  sin
.
.
.
,
a
x
d3
(6 V
(w)
=
1
:
sin
tr
&)),
cos
&lt;r
er
Q sin (9
.
+
x
&lt;).
*Archivfor Mathematik og Naturvidenskab, Vol. IV (1879), t Cf. Bianchi, Vol. II, p. 434; Darboux, Vol. Ill, p. 432.
p. 150.
290
.
SUEFACES OF CONSTANT CUEVATUEE
In particular, for a transformation of Bianchi we have
(B 4ccc
o))
= sin (B
o&gt;),
dp
(6
&&gt;)
= sin (6 +
&&gt;
o&gt;).
Suppose that we have a pair of functions 6 and satisfying these equations, and that we effect upon them the Lie transforma tion for which has the value (1 + cos cr)/sin a. This gives
m
1 4 cos
_
+
a
a
1
,
cos
sin
cr
cr
sin
or
/1
\
s s in
cr
sm cr
As
Ba
.
these functions satisfy (44), they determine a transformation But O l may be obtained from o^ by effecting upon the latter
1
Z" ,
an inverse Lie transformation, denoted by upon this result a B n/2 and then a direct Lie transformation, Bianchi transformation,
,
Z a Hence we may
.
write symbolically
which may be expressed thus
:
Backlund transformation B is the transform of a Bianchi * transformation ly means of a Lie transformation L a
ff
A
EXAMPLES
1.
The asymptotic
lines
on a pseudospherical surface are curves of constant
lines are of the
1.
torsion.
2.
Every surface whose asymptotic
is
same length as
the curves
their spherical
images
3.
a pseudospherical surface of curvature
that on the pseudosphere, defined
Show
by
(9),
=
where
4.
0,
6
is
a constant, are geodesies, and find the radius of curvature of these curves.
the linear element of a pseudospherical surface
is
When
(iii)
in the parabolic
form
of (7), the surface defined
by
y
a
dy
z z
x
is
=
x
.
a
76)
dx
du
pseudospherical (cf
*
;
y
it is
a
dz
cu
du
a Bianchi transform of the given surface.
The
Spherical surfaces admit of transformations similar to those of Lie and Backlund. such combinations of them can be made that the resulting surface is real. For a complete discussion of these the reader is referred to chap. v. of the Lezioni of Bianchi.
latter are imaginary, but
TFSURFACES
5.
291
The
X
helicoids
= U COS V.
fc
y
=
u
sin
u,
z= f */ J
\a
Idu
k~u z
u2
+
hv,
where
6.
a, A,
are constants, are spherical surfaces.
helicoid whose meridian curve is the tractrix is called the surface of Dini. equations when sin denotes the helicoidal parameter and cos the con stant length of the segment of the tangent between the curve and its axis. Show that the surface is pseudospherical.
The
Find
its
&lt;r
&lt;r
7. The curves tangent to the joins of corresponding points on a pseudospherical surface and on a Backlund transform are geodesies only when = ir/2.
&lt;r
8.
Let S be a pseudospherical surface and Si a Bianchi transform by means of
(
a function d
119).
Show
that
X{
cosw(cos0X1
f f
sin0JT2 ) sin0JT2 )
sinwJT,
X X
where
.Xi,
2
,
%
= =
sin
w
(cos&lt;?
Xi
+
coswJT,
X X are directioncosines, with respect to the xaxis,
on S and of the normal to
S,
to the lines of curvature
and
JT{,
XX
of the tangents are the %,
similar functions for Si.
123.
Wsurfaces.
and surfaces
great
Fundamental quantities. Minimal surfaces of constant curvature possess, in common with a
many
cipal radii is a function of the other.
first
other surfaces, the property that each of the prin Surfaces of this kind were
studied in detail by Weingarten, * and, in consequence, are called Weingarten surfaces, or simply Wsurfaces. Since the prin
cipal radii of surfaces of revolution
and of the general helicoids
are functions of a single parameter ( 46, 62), these are TFsurfaces. shall find other surfaces of this kind, but now we consider
We
the properties which are common to TFsurfaces. When a surface S is referred to its lines of curvature, the Codazzi equations may be given the form
(45)
glogV^ =
dv
1
dp^
Pi dv
P2
d
If a relation exists
between p l and
/&gt;
2,
as
the integration of equations (45)
r dpi
is
reducible to quadratures, thus
:
=Ue J **,
Crelle, Vol.
V^=
Ve
r J Pl
&lt;/p
2
~ P2
,
LXII
(1863), pp. 160173.
292
JFSUKFACES
and V are functions of u and v respectively. Without changing the parametric lines the parameters can be so chosen that the above expressions reduce to
where
U
/
A T\
re
and
r dp,
I
a
o
r^
f_dpj
I
expressible as functions of p l or /3 2 , and conse are functions of one another. This relation becomes quently they more clear when we introduce an additional parameter K defined by
(48)
Thus
^ are
*
=*
/
&lt;*pi
**
we have
a
By
the elimination of p 2 from this equation and (46)
j\(
\
relation of the form
When
this value is substituted in (48)
we
obtain
where the accent indicates differentiation with respect
(47)
it
to K.
From
follows that
.
V^=, K
,
^=T,
&lt;/&gt;
When
these values are substituted in the Gauss equation for
the sphere (V, 24), the latter becomes
1/*
du \
&lt;/&gt;"
M
du)
,
jL/*
dv
\tc*
aY
dv)
_1. =
K&lt;f&gt;
but
This equation places a restriction upon the forms of K and it is the only restriction, for the Codazzi equations (45) are
&lt;(),
satisfied.
Hence we have
the theorem of Weingarten
*
:
When
one has an orthogonal system on the unit sphere for which
the linear element is reducible to the
form
there exists a Wsurface whose lines of curvature are represented by
this
system and whose principal radii are expressed by
ft
(50)
=*(*),
P2
= *(*) * (*)
Z.c., p. 163.
FUNDAMENTAL QUANTITIES
If the
293
functions of
coordinates of the sphere, namely X, Y, Z, are known u and v, the determination of the JFsurface with this
For, from the formulas
representation reduces to quadratures.
of Kodrigues (IV, 32)
we have
x
=
=
r
dX
Pi
du
du
7
,
J
y
/
cu
+p +
dX
2
,
dv,
dv
C
pi
dY
,
dY
/? 2
,
^r~ dv,
cz = ~ rft ~^~ ^ + P* a^ dv
,

J
/
du
v"
dv
The righthand members
of these equations are exact differentials,
A",
since the Codazzi equations (45) have been satisfied. If F, Z are not known, their determination requires the solution of a
Riccati equation.
is
The
relation
between the
radii of the
form
(46)
obtained by eliminating K from equations (50). find readily that the fundamental quantities for the sur
We
face have the values
(51)
And from
(52)
(48), (50),
and (51) we obtain
t
&lt;
pi
Ve = p^
"
ft
,
vG = p,e
_
r
Jf
Pi
p
&gt;.
Consider the quadratic form
(53)
H [(EJJ FD) du
.
1
+ (El)" GD) dudv + (FD" GD
is
)
dv*],
which when equated
to zero defines the lines of curvature.
When
these lines are parametric, this quadratic form means of (IV, 74) to
reducible by
But
consequence of (47) this is further reducible for JFsurfaces to Since the curvature of this latter form is zero, the curvature of (53) also is zero, and consequently ( 135) the form (53) is redu cible by quadratures to dudv. Hence we have the theorem of Lie
in
dudv.
:
The
lines of curvature of
a Wsurface can be found by quadratures.
294
JFSURFACES
The evolute
of a JFsurface pos
124. Evolute of a Wsurface.
sesses several properties
results of
which are characteristic. Referring to the 75, by means of (52) the linear elements of the sheets of the evolute of a JFsurface are reducible to the form
we
see that
or, in
terms of
K,
(55)
From
these results and the remarks of
:
46 we obtain
at once
the following theorem of Weingarten
Each surface of center of a Wsurface is applicable to a surface of revolution whose meridian curve is determined by the relation between
the radii of the given surface.
We have also the converse theorem, likewise due to Weingarten
:
If a surface Sl be applicable to a surface of revolution, the tan meridians of the gents to the geodesies on S^ corresponding to the
surface of revolution are normal to a family of parallel Wsurfaces; the relation between the if Sl be deformed in any manner whatever,
radii of these Wsurfaces
is
unaltered.
In proving this theorem linear element of Sl be
i
we apply
i***
the results of
r
?
t
76.
If
the
the principal radii of
/tM
.
S
are given
by
V
p^u,
(56)
ft^7U
alone,
Since both are functions of a single parameter, a relation exists
between them which depends upon
unaltered in the deformation of
and consequently
is
Sr
trihedral for 8^ (V, 99) the projections upon the moving of a displacement of a point on the complementary surface 2 are
From
(___),
0,
(qdu,
ai
U ~ /ir
EVOLUTE OF A JFSUBFACE
is
295
In consequence of formulas (V, 48, 75) the expression U(q du + q t dv) an exact differential, which will be denoted by dw. Hence the
2
linear element of
(57)
is
dl =
it
l
from which
revolution.*
follows that
Sz
also is applicable to a surface of
The
last
theorem of
75
may
be stated thus
:
A necessary
and
sufficient condition that the
asymptotic lines on
correspond is that S be a Wsurface ; in this case to every conjugate system on Sl or S2 there corresponds a conjugate system on the other.
the surfaces of center
S^ S2
of a surface
S
From
(58)
.
(V, 98, 98
)
it
follows that
when S
is
a TFsurface, and
only in this case,
we have
.
.
^E^bsame kind.
is
Hence
at corresponding points the curvature is of the
afforded by the case where (46) one or both of the principal radii is constant. For the plane both radii are infinite for a circular cylinder one is infinite and the other
;
An exceptional form of equation
has a
finite
constant value.
The sphere
if
is
the only surface with both
For, p r and p 2 are different constants, from (45) it follows that and ^ are functions of u and v respec tively, which is true only of developable surfaces. When one of the radii is infinite, the surface is developable. There remains the case
radii finite
and constant.
where one has a finite constant value ; then S is a canal surface
In considering the last case
then, from (48),
(
29).
we take
we have
is
and the linear element of the sphere
do*
=
~+
K
dv\
Conversely, when the linear element of the sphere is reducible to this form, the curves on the sphere represent the lines of curvature
on an infinity of parallel canal surfaces.
* Cf. Darboux, Vol.
Ill, p. 329.
296
TFSURFACES
mean curvature. For
surfaces of con
125. Surfaces of constant
stant total curvature the relation (46)
may
be written
where
c
denotes a constant.
When this value
is
substituted in (48)
we
have, by integration,
(59)
P
is
so that the linear element of the sphere
(60)
Conversely,
when we have an orthogonal system on
is
which the linear element
the sphere for reducible to the form (60), it serves for
the representation of the lines of curvature of a surface of constant
curvature, and of an infinity of parallel surfaces. When c is positive, two of these parallel surfaces have constant
mean
curvature, as follows from the theorem of Bonnet fact, the radii of these surfaces tifcT
(61)
If
(73). In
pl
=^/ K *+cy/~c
J
p9 =
=L==
vK
~r~
V~ C
.
c
we put
c
(62)
=a
2
,
ic
=a
csch
&&gt;,
and replace u by
au, the linear
da
element (60) becomes
2
= sinlr co du +
cosh
2
o&gt;
dv
2
.
r
In like manner,
(63)
if
we
c
replace u by iau, v by
iv,
and take
=
a
2
,
K
=
2
ai sech
CD,
the linear element of the sphere
da2 2
is
= cosh w du +
we
sinh
2
&)
dv*.
For the values
(62)
have, from (61),
and the linear elements
a
of the corresponding surfaces are
a
(65)
&lt;f*
=ffV aw (dw
SURFACES OF CONSTANT MEAN CURVATURE
Moreover, for the values (63) the radii have the values
297
cosh
co
sinh
(65).
co
but the linear elements are the same
curvature
is
In each case the
mean
l/.
We
state these results in the following form:
upon a surface of constant mean curvature an isothermic system, the parameters of which can be chosen form so that the linear element has one of the forms (65), where co is a
The
lines of curvature
solution of the equation
l
(67)
du
^ 2
4
dv
^ 42
sinh
co
cosh
co
=
0.
Conversely, each solution of this equation determines two pair* of l/a, whose lines applicable surfaces of constant mean curvature
of curvature correspond, and for which the radii p^ p 2 of one surface are equal to the radii of p 2 p^ of the applicable surface.
,
It
can be shown that
is
co
if co
= $(u,
v sin
cr,
v) is
a solution of equation (67),
so also
(68)
1
= cf)(u cos
cr
u sin
&lt;r
+
v cos
cr),
where cr is any constant whatever. Hence there exists for spherical surfaces a transformation analogous to the Lie transformation of pseudospherical surfaces. This transformation can be given a geo
metrical interpretation if it is considered in connection with the sur faces of constant mean curvature parallel to the spherical surfaces.
Let Sl denote the surface with the linear element
(69)
If
ds 2 =
aV w
cr,
2
&gt;
(du
+ dv
=u
2
).
we put
u
v
(70)
=u cos
cr
v sin
co
l
v1
l,
sin
cr
+
v cos
cr,
the solution (68) becomes
= cf)(u
v^),
and
(69) reduces to
if we make a point (u, v) on S with the linear element (65), which the positive sign is taken, correspond to the point (u v vj on 8^ the surfaces are applicable, and to the lines of curvature u = const., v = const, on S correspond on Sl the curves
Hence
in
u cos
cr
v sin
a
= const.,
u
sin
cr f
v cos
cr
= const.
298
^SURFACES
latter cut the lines of curvature
But the
l
u
=
const., v
= const,
on
S under the angle a. Moreover, the corresponding principal radii of S and S are equal at corresponding points. Hence we have tha
l
* following theorem of Bonnet
:
A
surface of constant
mean curvature admits an
and
infinity of appli
cable surfaces of the
same kind with preservation of
the principal
radii at corresponding points,
the lines of curvature on one
the
lires
surface correspond to lines on the other which cut curvature under constant angle.
of
of Weingarten has considered the IFsurfaces whose lines curvature are represented on the sphere by geodesic ellipses and hyperbolas. In this case the linear element of the sphere is reducible to the form ( 90)
do* =
sm
Comparing
this
CS
*2
*2
with
/c
(49),
we have
.,
&lt;
=
.to
sin&gt;
.
=cosft)
ft)
from which
it
follows that
to f
sin
4
Hence
&)
f sin
ft)
ft)
sin
and the
relation
between the
radii is found,
by the elimination
of w, to be
(72)
2(^^)=sin2(^+/)
,
2
).t
* Memoire sur la theorie des surfaces applicables sur une surface donnce, Journal de solves com VEcole Poly technique, Cahier 42 (1867) pp. 72 et seq. In this memoir Bonnet surfaces with corresponding principal radii equal. pletely the problem of finding applicable When a surface possesses an infinity of applicable surfaces of this kind, its lines of curv ature form an isothermal system. follows: tDarboux (Vol. Ill, p. 373) proves that these surfaces may be generated as locus of the Let C and Ci be two curves of constant torsion, differing only in sign. The of translation. of the join of any points P and PI of these curves is a surface midpoints of the osculating planes of C and If a line be drawn through parallel to the intersection above type for all positions of M. Ci at P and Pi, this line is normal to a IPsurface of the
M
M
RULED JFSURFACES
126.
299
Ruled Wsurfaces.
We
conclude the present study of
Trsurfaces with the solution of the
problem
:
To determine
the Wsurfaces
which are ruled.
This problem was proposed and solved simultaneously by Beltrami* and Dini.f We follow the method of the latter. In 106, 107 we found that when the linear element of a
ruled surface
is
in the
form
2
ds
2
= du + [(u  a) + /3
2
2
]
dv\
are
the expressions for the total and
2
mean curvatures
~
where r
is
/3
=
a function of v at most, and
/=(tt~a) h^.
In order that a relation exist between the principal radii
necessary and
sufficient that the equation
it is
a
1* a* du
dv
dv
*jr.l*:o du
above values be substituted, the
be satisfied identically.
If the
resulting equation reduces to
2u
a
d rr
2
+/3
!
ua
a
l
\
As
this is
it
case
an identical equation, reduces to /3 =0. Hence
r (u
it is
/3 is
2
true
when u =
a, in
which
a constant and the above
equation becomes
of +
r
ft
+
/3a"
=
0.
Since this equation must be true independently of the value of w, both r and are zero. Therefore we have
a"
(73)
a=cv +
&lt;?,
d,
P=
e,
r
= k,
where
d, e,
k are constants.
is
The
linear element
ds
*
2
=
du2 +
[(t*
 cv  d) + e
2 t
2
]
dv
2
.
Annali, Vol. VII (1865), pp. 13&150.
Annali, Vol. VII (1865), pp. 205210.
300
SURFACES WITH PLANE LINES OF CURVATURE
for
In order to interpret this result we calculate the expression the tangent of the angle which the generators v = const, make with the line of striction
u
cv
d
=
0.
From
(III, 24)
we have
tan d
=c
;
6 the param consequently the angle is constant. Conversely, if and eter of distribution j3 be constant, a has the form (73). Hence we
have the theorem
:
be a necessary and sufficient condition that a ruled surface distribution be constant and that is that the of Wsurface
A
parameter
is
the generators be inclined at a constant angle to the line of stric
tion,
which consequently
a geodesic.
EXAMPLES
1.
Show
that the helicoids are ^surfaces.
2. Find the form of equation (49), when the surface is minimal, and show that each conformal representation of the sphere upon the plane determines a minimal
surface.
3.
Show
the linear element
that the tangents to the curves v = const, on a spherical surface with of (6) are normal to a TTsurface for which (i)
P2

PI
=
COt

4.
The
const,
helicoids
are the only
&gt;Fsurfaces
Pi
=
5.
meet the
lines of curvature
lines
which are such that the curves under constant angle (cf. Ex. 23, p. 188).
The asymptotic
Pz
;
Pl
+
const, correspond to the
of the surface
and,
when
/&gt;i
on the surfaces of center of a surface for which minimal lines on the spherical representation on the sphere. p 2 = const., to a rectangular system
127. Spherical
representation of surfaces with plane lines of
curvature in both systems. Surfaces whose lines of curvature in one or both systems are plane curves have been an object of study to a line of curvature and by many geometers. Since the tangents to its spherical representation at corresponding points are parallel, a plane line of curvature is represented on the sphere by a plane is plane curve, that is, a circle and conversely, a line of curvature
;
when
its
spherical representation
is
a circle.
SPHERICAL REPRESENTATION
301
We
lines
consider
first
the
of curvature in
determination of surfaces with plane both systems from the point of view of
their spherical representation.* To this end we must find orthog onal systems of circles on the sphere. If two circles cut one another orthogonally, the plane of each must pass through the
pole of the plane of the other. Hence the planes of the circles of one system pass through a point in the plane of each circle of the second system, and consequently the planes of each family
form a pencil, the two axes being polar reciprocal with respect to
the sphere.f consider separately the two cases I, when one axis is tan gent to the sphere, and therefore the other is tangent at the same
We
:
point and perpendicular to
it
;
II,
when
neither
is
tangent.
CASE
x
I.
We
and
?/axes parallel to the
take the center of the unit sphere for origin 0, the axes of the pencils, and let the coor
dinates of the point of contact be (0, 0, 1). pencils of planes may be put in the form
(74)
The equations
of the
x
v
+ u(z
are
1)=0,
y
+ v(z
1)
=
0,
where u and
the parameters of the respective families. If these equations be solved simultaneously with the equation 7 of the sphere, and, as usual, X, I Z denote coordinates of the
,
latter,
we have
v
Now
(T6)

^v
r7
_u?~
the linear element of the sphere
is
^=
JtXl?
CASE II. As in the preceding case, we take for the zaxis the common perpendicular to the axes of the pencils, and for the xand ?/axes we take lines through parallel to the axes of the
coordinates of the points of meeting of the latter with the zaxis are of the form (0, 0, a), (0, 0, I/a). The equa tions of the two pencils of planes could be written in forms
pencils.
*
t
The
Bianchi, Vol. II, p. 256; Darboux, Vol. I, p. 128, and Vol. IV, p. 180. (1853), pp. 136, 137. Bonnet, Journal de I Ecole Poly technique, Vol.
XX
302
SURFACES WITH PLANE LINES OF CURVATURE
similar to (74), but the expressions for X, Y, Z will be found to be of a more suitable form if the equations of the families
of planes be written
tanw
atanhv
Proceeding as in Case
I,
we
find
Vl
cosh v
(77)
a sin u
2
+
a cos u
a sinh v
f
2
1 Y= cosh v
a cos u
Z=
and the linear element
(78)
is
cos
u
\
a cosh v
cosh v
f
a cos w
(cosh v
+ a cos w)
we have tacitly excluded the sys and parallels. As before, the planes of the two families of circles form pencils, but now the axis of one pencil passes through the center of the sphere and the other is at infinity.
From
the preceding discussion
tem
of meridians
Hence
fact, if
this case
corresponds to the value zero for a in Case
II.
In
we put
a
=
referred to a system
_
(
in (77), the resulting equations define a sphere of meridians and parallels, namely
JL
I
Q V
)
JL
sinw

t
sinhv  
&gt;
Z/
cosw

cosh v
cosh v
cosh v
Since the planes of the lines of curvature on a surface are parallel const, on a to the planes of their spherical images, the curves v surface with the representation (79) lie in parallel planes, and the
planes of the curves u = const, envelop a cylinder. These surfaces shall consider them later. are called the molding surfaces.*
We
128. Surfaces
with plane
choice
of
lines of curvature in both systems.
By
a
suitable
coordinate
axes and parameters the
expressions for the directioncosines of the normal to a surface with plane lines of curvature in both systems can be given one
* These surfaces were first studied
trie,
by Monge, Application de
L
Analyse a la Geomt
17.
Paris, 1849.
IN
of the forms (75) or (77).
BOTH SYSTEMS
303
surfaces of this kind
it
For the complete determination of all remains then for us to find the expres
sion for the other tangential coordinate W, that is, the distance from the origin to the tangent plane. The linear element of the sphere in both cases is of the form
d(T =
7
2
g
du 2 + dv 2 &gt;
where \
(80)
is
such that
^ = 0.
cudv
(VI, 39)
From
we
see that the equation satisfied
gfl
by
W
is
cucv
dv
du
log du
X d6
dv
_Q
In consequence of (80), if we change the unknown function in accordance with B l =\0 the equation in 6 l is of the form (80).
&gt;
Hence the most general value
*for
W
is
where U and V are arbitrary functions of u and v respectively. Hence any surface with plane lines of curvature in both systems is the envelope of a family of planes whose equation is of the form
(81)
2 ux
+
2 vy
+ (u*+ v l)z = 2 (U+V),
2
or
(82)
Vl
a"
sin
ux
Vl
a sinh vy
2
.
2
+ (cos u + a cosh v) z
= (U+ F)Vla
The
expressions for the Cartesian coordinates of these surfaces 67. Thus, can be found without quadrature by the methods of
for the surface envelope of (81)
we have
to solve for x, y, z
equa
tion (81)
(83)
and
its
derivatives with respect to
u and
v.
The
latter are
x
+ uz =
Z7
,
y
+
vz
= V\
where the accents indicate differentiation. We shall not carry out this solution, but remark that as each of these equations contains
a single parameter they define the planes of the lines of curvature.
304
SURFACES WITH PLANE LINES OF CURVATURE
the form of (83)
it is
From
seen that these planes in each sys
tem envelop a
these two cylin cylinder, and that the axes of This fact was remarked by Darboux, ders are perpendicular.
who
also observed that equation
(81) defines the radical plane
of the
two spheres
These are the equations whose centers lie on the
of
two oneparameter families
of spheres,
focal parabolas
U,
and whose
radii are
2/1=0,
determined by the arbitrary functions
its
U and
V.
The characteristics of each famity are defined by the corresponding equation of the pair (83). Consequently the orig inal surface is the locus of the point of intersection of the planes of these characteristics and the radical planes of the spheres.
Similar results follow for the equation (82), which defines the radical planes of two families of spheres whose centers are on the
focal ellipse
equation and
and hyperbola
(86)
a;
2
=0,
2/ 2
= =
0,
When
in particular a
these curves of center are a circle and
r
i
its axis.
From
the foregoing results
it
follows that these surfaces
to
may
be
:
generated by the following geometrical method due
Darboux
*
be Every surface with plane lines of curvature in two systems can obtained from two singly infinite families of spheres whose centers lie on focal conies and whose radii vary according to an arbitrary law.
The surface
belonging
infinitely
to
is the
S and 2, envelope of the radical plane of two spheres two different families. If one associate with S and 2 two S and 2
f ,
near spheres
the radical center of these
the radical
four
spheres describes the surface ; and of 2 and 2 are the planes of the lines of curvature.
* Vol.
i,
and planes of S and S
p. 132.
SURFACES OF MONGE
129. Surfaces with plane lines
of
305
curvature in one system.
the lines of curvature in one system Surfaces of Monge. are plane, the curves on the sphere are a family of circles and
this
When
and conversely. Every system of ; be obtained from a system of circles and their may orthogonal trajectories in a plane by a stereographic projection.
their orthogonal trajectories
kind
The determination
of such a system in the plane reduces to the
integration of a Riccati equation (Ex. 11, p. 50). Since the circles are curves of constant geodesic curvature we have, in consequence
of the first theorem of
84, the
all the
theorem
:
The determination of
quadratures.
surfaces with plane lines of curva
ture in one system requires the solution of a Riccati equation
and
lines of curvature in
with plane one system, and begin with the case where these curves are geodesies. They are consequently normal sections
of the surface.
We
shall discuss at length several kinds of surfaces
Their planes envelop a developable surface, called the directordevelopable, and the lines of curvature in the other sys
tem
are the orthogonal trajectories of these planes. Conversely, the locus of any simple infinity of the orthogonal trajectories of a oneparameter system of planes is a surface of the kind sought.
For, the planes cut the surface orthogonally, and consequently they are lines of curvature and geodesies ( 59). Since these
planes are the osculating planes of the edge of regression of the developable, the orthogonal trajectories can be found by
quadratures
(
17).
Suppose that we have such a surface, and that C denotes one of
the orthogonal trajectories of the family of plane lines of curvature. Let the coordinates of C be expressed in terms of the arc of the
curve from a point of it, which will be denoted by v plane of each plane line of curvature F is normal to C at
.
As
its
the
point
of
meeting with the
latter,
reference to the
moving
the coordinates of a point trihedral of C are 0, 77, f. Since
P of F with P describes
82)
an orthogonal trajectory of the planes, we must have
(I,
dv
306
SURFACES WITH PLANE LINES OF CURVATURE
C.
where r denotes the radius of torsion of
parameter of
If
we change
the
C
in accordance with the equation
the above equations become
The
(88)
general integral of these equations
?;
is
= U^ cos v
l
U
2
sin v^
f
= U^ sin v
l
f ?72
cos v lt
where C^ and
?72 are
functions of the parameter u of points of F.
l
When v = we have v = 0, and so the curve F in the plane through = U^ ?= Z7 Hence the of C has the equations the point v = character of the functions U^ and U is determined by the form of the curve and conversely, the functions U and U determine the
77
2
.
2
;
}
2
character of the curve.
By
definition
(87)
out in the plane normal to
the function v t measures the angle swept C by the binormal of the latter, as this
= to any other point. Hence equations (88) plane moves from v define the same curve, in this moving plane, for each value of v^
but
it is
the angle v r
defined with respect to axes which have rotated through Hence we have the theorem
:
surface whose lines of curvature in one system are geodesies can be generated by a plane curve whose plane rolls, without slipping,
Any
over a developable surface.
These surfaces are called the surfaces of Monge, by whom they were first studied. He proposed the problem of finding a surface
with one sheet of the
e volute a developable. It is evident that the
above surfaces satisfy this condition.
only solution. ment lie in the plane tangent along this element, and if these tangents are normals to a surface, the latter is cut normally by this plane, and consequently the curve of intersection is a line of
curvature.
of
Moreover, they furnish the the tangents to a developable along an ele For,
Monge
In particular, a molding surface ( 127) with a cylindrical directordevelopable.
is
a surface
Since every curve in the moving plane of the lines of curva ture generates a surface of Monge, a straight line in this plane
MOLDING SURFACES
307
generates a developable surface of Monge. For, all the normals to the surface along a generator lie in a plane ( 25). Hence:
necessary and sufficient condition that a curve F in a plane normal to a curve C at a point Q generate a surface of Monge as
A
plane moves, remaining normal to the curve, is that the joining a point of T to Q generate a developable.
the
.
line
the orthogonal trajectory C is a curves F are perpendicular to the plane curve, the planes of the plane of C, and consequently the directordevelopable is a cylinder whose right section is the plane evolute of C. The surface is a
130. Molding surfaces.
When
molding surface
(
127),
and
all
the lines of curvature of the sec
ond system
involutes of the right section of the cylinder. Hence a molding surface may be generated by a plane curve whose plane rolls without slipping over a cylinder.
are plane curves,
We
shall apply the preceding formulas to this particular case. are Since 1/r is equal to zero, it follows from (88) that ?; and
If
all
functions of u alone.
u be taken
as a measure of the arc of the
curve F, we have, in
?;
generality,
=
U
C
U,
f
=
I
Vl
U
2
du,
If
where the function
plane of the curve
determines the form of F.
for 2
we
take the
=
0,
and XQ yQ denote the coordinates
,
of a point of C, the equations of the surface
may
2
be written
/
x
= x + U cos v,
Since
y
=
2/o
+ u sin v
^
i
=
Vl
to
U
2
du,
where v denotes the angle which the principal normal
the
a&gt;axis.
C makes with
^x^
= sin v
(
,
=
C,
cos
v,
if
V
denote the radius of curvature of
*
:
then ds Q
= V dv,
and the
equations of the surface can be put in the following form, given by
Darboux
(
r
v
fI
Fsin
J
I
v dv,
(89)
= U sin v
V cos v dv,
* Vol.
I, p.
105.
308
SURFACES WITH PLANE LINES OF CURVATURE
of the right section of the cylinder are
The equations
x
= X + V cos v =
Q
I
V cos v dv,
V
sin v dv.
y
In passing,
faces,
=y
Q f
V sin v =
I
we remark
whose
0.
that surfaces of revolution are molding sur this corresponds to the directorcylinder is a line
;
case
V
EXAMPLES
a surface is 1. When the spherical representation of the lines of curvature of isothermal and the curves in one family on the sphere are circles, the curves in the other family also are circles.
2.
If the lines of curvature in
one system on a minimal surface are plane, those
in the other
3.
system also are plane.
that the surface
__
Show
x
au
sin
u cosh v,
its lines
y
=
v
+
a cos u sinh v,
z
V
1
a 2 cos u cosh
v,
is
minimal and that
of curvature are plane.
Find the spherical representa
tion of these curves
4.
and determine the form of the curves.
Show
that the surface of Ex. 3 and the Enneper surface (Ex. 18, p. 209) are
the only minimal surfaces with plane lines of curvature.
5.
is
When
the lines of curvature in one system
lie
in parallel planes, the surface
of the
molding type.
6. A necessary and sufficient condition that the lines of curvature in one system on a surface be represented on the unit sphere by great circles is that it be a sur
face of Monge.
7.
Derive the expressions for the point coordinates of a molding surface by the
of
67.
method
131. Surfaces of Joachimsthal. Another interesting class of surfaces with plane lines of curvature in one system are those for which all the planes pass through a straight line. Let one of these lines of curvature be denoted by F, and one of the other system
by
C.
The developable enveloping
the surface along the latter has
for its elements the tangents to the curves F at their points of intersection with 0. Since these elements lie in the planes of the
curves F, the developable is a cone with its vertex on the line to the through which all these planes pass. This cone is tangent surface along (7, and its elements are orthogonal to the latter. Con
Z&gt;,
sequently
C
is
the intersection of the surface and a sphere with
SURFACES OF JOACHIMSTHAL
309
center at the vertex of the cone which cuts the surface orthogo * nally. Hence we have the following result, due to Joachimsthal
:
the lines of curvature in one system lie in planes passing a line D, the lines of curvature in the second system lie on through and which cut the surface orthogonally. spheres whose centers are on
When
D
Such surfaces
curves of the
are called surfaces of Joachimsthal.
Each
of the
circles
first
system
is
an orthogonal trajectory of the
in which the spheres are cut by its plane. Therefore, in order to derive the equations of such a surface, we consider first the orthog onal trajectories of a family of circles whose centers are on a line.
If the latter be
taken for the
f
?;axis,
the circles are defined by
= r sin 0,
77
= r cos 6 + u,
where r denotes the radius, 6 the angle which the latter makes with the ?;axis, and u the distance of the center from the origin. Now r is a function of u, and 6 is independent of u. In order that these same equations may define an orthogonal trajectory of the circles, 6 must be such a function of u that
cos
or
r
0^ sin 0^ =
du cu
0,
f^_ sin = du
tan
.
By
integration
we have
(90)
=
r
F&lt;/
,
where
V denotes
the constant of integration.
Since each section of a surface of Joachimsthal by a plane
through its axis is an orthogonal trajectory of a family of circles whose centers are on this axis, the equations of the most general
surface of this kind are of the form
x
= r sin 6 cos v,
y
= r sin 6 sin v,
z
u
f r cos #,
where
v denotes the angle
the axis makes with the plane which now V is a function of v.
*
Crelle, Vol.
which the plane through a point and y 0, and 6 is given by (90), in
LIV
(1857), pp. 181192.
310
SURFACES WITH PLANE LINES OF CURVATURE
constant
is
When V is
is
a function of u alone, and the surface
For other forms of Vihe geometrical genera tion of the surfaces is given by the theorem
one of revolution.
:
Given
ters lie
the orthogonal trajectories of a
line
family of
circles
whose cen
on a right
D;
to
ferent angles, according surface of Joachimsthal.
through dif a given law, the locus of the curves is a
if they be rotated about
D
132. Surfaces with circular lines of curvature.
We
consider
next surfaces
whose
lines of curvature in one
Let
o
denote the constant angle between
system are circles. the plane of the circle
the tangent planes to the surface along C (cf. 59), p the radius of normal curvature in the direction of C, and r the radius
C and
of the latter.
(91)
Now
equation (IV, 17)
r
may
be written
= p sin a.
the theorem
:
As an immediate consequence we have
A necessary
be the
and
all
sufficient condition that
be a circle is that the
a plane line of curvature normal curvature of the surface in its direction
points.
same at
of
its
Since the normals to the surface along C are inclined to its plane under constant angle, they form a right circular cone whose vertex is on the axis of C. Moreover, the cone cuts the surface at right and center at the angles, and consequently the sphere of radius p
vertex of the cone
surface
is
is
tangent to the surface along C.
Hence the
the envelope of a family of spheres pf variable or con stant radius, whose centers lie on a curve.
Conversely, we have seen in the family of spheres
29 that the characteristics of
where x, y, z are the coordinates of a curve expressed in terms of its of radius arc, and 11 is a function of the same parameter, are circles
(92)
r
whose axes are tangent
have the coordinates
(93)
to the curve of centers
and whose centers
xl = x 
aRR
,
y
CIRCULAR LIKES OF CURVATURE
where
a,
ft,
311
indicates differentiation.
characteristic
7 are the directioncosines of the axis, and the accent The normals to the envelope along a
form a cone, and consequently these
circles are lines
of curvature
upon
and
it.
Hence
:
of curvature in one family be circles is that the surface be the envelope of a single infinity of spheres, the locus of whose centers is a curve, the radii
being determined by an arbitrary law.
A necessary
sufficient condition that the lines
From
surfaces
equations (91), (92)
it
follows that
R
cos a.
is,
Hence the
for canal
circles are geodesies only
(
when
R
is
is
constant, that
29).
In this case, as
seen from (92),
all
the circles
are equal.
The
circles are likewise of equal radius a
when
where
s is
the arc of the curve of centers and c
is
a constant of
integration.
Now
(s
equations (93)
a.
become
(s
^=x
(
+
c)
y l =^y
also of
+ c}IB,
:
zl
=z
(s
+ c)y,
which are the equations
21).
an involute of the curve of centers
This result
be
may
be stated thus*
extremity M generates
be
If a string
a curve in such a way that its moving a circle an involute of the curve, and if at constructed whose center is and whose plane is normal to the
unwound from
M
M
string, then as the string is unwound this circle generates a surface with a family of equal circles for lines of curvature.
locus of the centers of the spheres enveloped by a surface is evidently one sheet of the evolute of the surface, and the radius
of the sphere is the radius of
of
The
normal curvature in the direction
this
the
circle.
Consequently
radius
is
a
function
75,
of
the
that
parameter of the spheres.
when
2
is
a curve
H=
2
0,
Conversely, from and consequently
we have
Cf. Bianchi, Vol. II, p. 272.
312
SURFACES WITH PLANE LINES OF CURVATURE
Excluding the case of the sphere, we have that p.2 is a function of u alone. From the formulas of Rodrigues (IV, 32),
dX _ d^~~~ Pz ~^
2x
dy
1
_
dY
dz
fo~~ p2 ~fo
Tv~~ pz ^v
_
a
we
have, by integration,
Hence the points
of the surface lie
a 2 )
on the spheres
,
 U, + (y  tg + (z  P,) = ft (x
and the spheres are tangent
to the surface.
Since the normals to a surface along a circular line of curvature form a cone of revolution, the second sheet of the e volute is the
envelope of a family of such cones. The characteristics of such a family are conies. Hence we have the theorem
:
necessary and sufficient condition that one sheet of the evolute of a surface be a curve is that the surface be the envelope of a single infinity of spheres ; the second focal sheet is the locus of a family of conies.
133. Cyclides of Dupin.
A
that
it is
if
the preceding theorem it results also the second sheet of the evolute of a surface be a curve,
From
a conic,
and then the
first
sheet also
is
a conic.
Moreover, these
conies are so placed that the cone formed by joining any point on one conic to all the points of the other is a cone of revolution. pair of focal conies is characterized by this property. And so
A
we have
the theorem
:
A
both families be circles
necessary and sufficient condition that the lines of curvature in is that the sheets of the evolute be a pair of
focal conies.*
These surfaces are called the cy elides of Dupin. They are the envelopes of two oneparameter families of spheres, and all such
sphere of one family touches envelopes are cyclides of Dupin. each sphere of the other family. Consequently the spheres of which the cyclide is the envelope are tangent to three spheres.
A
We
is
shall prove the converse
theorem of Dupin f
to
:
The envelope of a family of spheres tangent
a cyclide.
* Cf. Ex. 19, p. 188.
t
three fixed spheres
Applications de geomttrie
et
de mechanique, pp. 200210.
Paris, 1822.
CYCLIDES OF DUPIN
313
The plane determined by the centers of the three spheres cuts the latter in three circles. If any point on the circumference (7, orthogonal to these circles, be taken for the pole of a transforma
tion
by reciprocal
radii (cf.
80),
C
is
transformed into a straight
line L.
Since angles are preserved in this transformation, the three fixed spheres are changed into three spheres whose centers are on L. Evidently the envelope of a family of spheres tangent to these three
is
spheres
a tore with
tore.
L
as axis.
Hence the given envelope
is
trans
However, the latter surface is the envelope of a second family of spheres whose centers lie on L. Therefore, if the above transformation be reversed, we have a second family of spheres tangent to the envelope, and so the latter is a cyclide of
Dupin. We shall now find the equations of these surfaces. Let (x^ y^ zj and (#2 y 2 z 2 ) denote the coordinates of the points on the focal conies which are the curves of centers of the
, ,
formed into a
and
(94)
jR 1?
E
2
the radii of the spheres.
( Xi
The
spheres, condition of tangency is
x
the case where the evolute curves are the focal
We
consider
first
parabolas defined by (85).
Now
equation (94) reduces to
Since
is
2i l
and
R
z
are functions of
u and
v respectively, this equation
equivalent to
where a
surfaces.
is
an arbitrary constant whose variation gives parallel
By the method of 132 we find that the coordinates (f, 77, f) of the centers of the circular lines of curvature u const, and the
radius p are
90,
314
SURFACES WITH PLANE LINES OF CURVATURE
if
be a point on the circle and 6 denote the angle which the radius to P makes with the positive direction of the normal to
Hence
P
the parabola (85), the coordinates of
P
are
2
x
=
fH
^
cos
2
v1 +u
is
0,
^
= p sin 0,
= ? 
cos
if
9.
Vl +
This surface
If
algebraic and of the third order. the evolute curves are the focal ellipse and hyperbola (86), we have
(96)
R =  (a cos u + *),
l
!
A =  (cosh v
2

/c),
where /c is an arbitrary constant whose variation gives parallel surfaces.
This cyclide of Dupin is of the fourth degree. When in particular the constanta is zero, the surface is the ordinary tore, or anchor ring.*
with spherical lines of curvature in one system. Surfaces with circular lines of curvature in one system belong evi dently to the general class of surfaces with spherical lines of curva
134. Surfaces
consider now surfaces of the latter kind. ture in one system. S be such a surface referred to its lines of curvature, and Let
const, be spherical. The coordinates in particular let the lines v of the centers of the spheres as well as their radii are functions of
v alone.
We
=
They
will be denoted
(
thal s
theorem
by (V^ F2 F3 ) and It. By Joachimseach sphere cuts the surface under the same 59)
,
angle at all its points. Hence for the family of spheres the expres sion for the angle is a function of v alone ; AVC call it V. Since the directioncosines of the tangent to a curve u = const, are
dX
when the
do2
3Y
1
dZ
=
(odu
linear element of the spherical representation 2 }dv\ the coordinates of S are of the form
is
written
R sin VdX = VA=+XR
,
,
cos F,
T_
/07\ (97)
y
=F +
2
Tr + YR
_
cos F,
7 sin
F dZ
*
to the article in the
For other geomotrical constructions of the cyclides of Dupin the reader is referred Encyklopadie der Math. Wissenschaflen, Vol. Ill, 3, p. 290.
SPHERICAL LINES OF CURVATURE
By
r hypothesis A, l
,
315
Z are
the directioncosines of the normal to
S
;
consequently we must have
YA = O, ^
du
If the values of the derivatives
VA = O. ^
dv
means
of (V, 22),
and the
obtained from (97) be reduced by results substituted in the above equa
tions, the first vanishes identically
and the second reduces
to
(98)
XV[ + YVt + ZVZ + (R
when
cos V)
R sin FvV = 0,
to
v.
where the primes indicate differentiation with respect
versely,
this condition is satisfied,
Con
:
equations (97) define a
surface on which the curves v
= const,
are spherical.
Hence
A necessary and
can
Ie
sufficient condition that the curves v
=
const, of
an
orthogonal system on vature upon a surface
the unit sphere represent spherical lines of
is
cur
R,
V,
found which
that five functions of v, namely Vr F2 , satisfy the corresponding equation (98).
F
3,
We
only
note that
FF
1?
2,
F
to within additive constants.
for the first three gives a
R cos V are determined by (98) A change of these constants translation of the surface. If R cos V be
3,
and
increased by a constant, other one. Hence *
:
we have
a
new
surface parallel to the
the
If the same
lines of curvature in one system
is
upon a surface
be spherical,
true of the corresponding system on each parallel surface.
,
Since equation (98) is homogeneous in the quantities F/, F^, F3 (R cos F) R sin F, the latter are determined only to within a factor which may be a function of v. This function may be chosen so
,
that all the spheres pass through a point. have the theorem of Dobriner f
:
From
these results
we
With each surface with spherical lines of curvature in one system there is associated an infinity of nonparallel surfaces of the same
kind with the same spherical representation of these lines of curvature. Among these surfaces there is at least one for which all the spheres
pass through a point. At corresponding points of the loci of the cen ters of spheres of two surfaces of the family the tangents are parallel.
* Cf. Bianchi, Vol.
II, p. 303.
f
Crelle, Vol.
XCIV
(1883), pp. 118, 125.
316
SURFACES WITH PLANE LINES OF CURVATURE
from
(97) be substituted in the formulas
If the values of x, y, z
of Rodrigues (IV, 32),
dx
&lt;
dx
"*
dx
99)
for
a^^
means
dx
of (V, 22),
and similarly
y and
z,
we
obtain by
=R cos
Conversely,
when
for a surface referred to its lines of curvature
is
the principal radius p l
of the
form
.*."
where (^ and $ 2 are any functions whatever v = const, are spherical. For, by (V, 22),
of
v,
the curves
dv
V~
ov
du
is
Consequently, from the
value,
first
of (99), in
which p l
given the above
we
obtain by integration
where
V
l
is
a function of v alone.
As
v=
these expressions are of the form (97),
Similar results follow for y and we have the theorem
z.
:
A
necessary and sufficient condition that the lines of curvature
be spherical is that p l be of the
const,
form
(100).
EXAMPLES
1.
If the lines of
curvature in one system are plane and one
is
a circle,
all
are circles.
2.
When
the lines of curvature in one family on a surface are circles, their
indispherical images are circles whose spherical centers constitute the spherical catrix of the tangents to the curve of centers of the spheres which are enveloped surface. Show also that each oneparameter system of circles on the the
by
given unit sphere represents the circular lines of curvature on an infinity of surfaces, for one of which the circles are equal.
EXAMPLES
3.
31T
If the lines of
curvature of a surface are parametric, and the curves u
j
=
const.
are spherical,
we have
j
cot
Pi
F
Pgu
B sin F
the radii of geodesic curvature and normal curvature in the denotes the angle under const, and of the sphere respectively, and direction v which the sphere cuts the surface.
where
pgu ,
/&gt;i,
E denote
F
4. When a line of curvature is spherical, the developable circumscribing the surface along this line of curvature also circumscribes a sphere and conversely, if such a developable circumscribes a sphere, the line of curvature lies on a sphere
;
concentric with the latter
5.
(cf.
Ex.
7, p. 149).
Let S be a pseudospherical surface with the spherical representation
of curvature.
its lines
Show
that a necessary
a/
1
and
(25) of sufficient condition that the curves
v
= const,
be plane
is
a&\
_
du \sin w dv/
show
also that in this case
w
is
given by
COS
0)
=
V U
,
where
V and V are functions of u and U * = + (a  2) C72 + 6,
4
U"
v respectively,
2
which
2
satisfy the conditions
(a f b
F = F* + aF
f

1),
a and
"V
b being constants,
and the accent indicating
differentiation, unless
U
or
is
zero.
6.
When
the lines of curvature v
is
const,
upon a pseudospherical surface are
plane, the linear element
reducible to the form
_ ~
where A, B,
7.
a 2 sech 2 (u 4 v} dv 2 a2 tanh 2 (u + v) dw2 C A cosh 2 u f B sinh 2 u G + A cosh 2 v + B sinh 2 v  1 C are constants. Find the expressions for the principal radii.
the lines of curvature v
is
When
=
v)
const,
on a spherical surface are plane, the
a2 esc 2 (u
linear element
reducible to
_ ~
where
J.
a 2 cot2 (u
+
dw2
I
+
v}
d
i?
2
.A sin 2 w f B 
A sin 2 u  7?
of Exs. 5
and
J5
are constants.
The surfaces
and 6 are called
the surfaces
of Enneper of constant curvature.
GENERAL EXAMPLES
1. The lines of curvature and the asymptotic lines on a surface of constant curvature can be found by quadratures.
2.
the equations x = cw, y upon the plane, which the plane
this axis.
When the linear element of a pseudospherical surface is in the form (iii) of (7), M = ae~a determine a conformal representation of the surface
is
by a
circle
with
such that any geodesic on the surface is represented on its center on the icaxis, or by a line perpendicular to
318
TFSURFACES
3. When the linear elements of a developable surface, a spherical surface, and a pseudospherical surface are in the respective forms ds 2 = a?(du 2 + sin 2 wdu 2 ), ds 2 = 2 (dw 2 + sinh^udw 2 ), ds~ = du 2 + udv 2
,
the finite equations of the geodesies are respectively
Au
cos v
f
Bu sin v
f
C=
0,
A
+
.B
tan u cos v
A
Z&gt;,
tanh u cos v
;
tanh u sin v
C are constants if the coefficients of where A, to x and y, the resulting equations define a correspondence between the surface and the plane such that geodesies on the former correspond to straight lines on the latter. Find the expression for each linear element in terms of x and y as parameters.
4.
B tan u sin v + C = 0, + C = 0, A and B are in any case equated
f
Each surf ace of center of a pseudospherical surf ace is applicable
to the catenoid.
5.
The asymptotic
lines
curvature correspond to
6.
on the surfaces of center of a surface of constant mean the minimal lines on the latter.
u and
if
=
7.
const.
Surfaces of constant mean curvature are characterized by the property that is a function of u alone v = const, are the minimal curves, then
,
D
D"
of v alone.
(23)
Equation
admits the solution w
=
0, in
which case the surface degen
erates into a curve.
(35) is tan 0/2
Show
M+
r cos
Bin&lt;r
that the general integral of the corresponding equations
&lt;r
=
Ce
;
take for
S
the line x
=
0,
y

0,
z
=
an and derive
the equations of the transforms of 122), or a pseudosphere. (Ex. C,
S;
shc^w that the latter are surfaces of Dini
8. Show that the Backlund transforms of the surfaces of Dini and of the pseudosphere can be found without integration, and that if the pseudosphere be trans formed by the transformation of Bianchi, the resulting surface may be defined by
x
=
2
a cosh u
V
2 a cosh
u
.
(sinu
ucosu),
y
sinh u cosh u\
(cosv
V
+ vsm v),
(2
U
cosh 2 w
+
v2 /
Show
9.
that the lines of curvature v
const, lie in planes through the 2axis.
The tangents
to a family of geodesies of the elliptic or hyperbolic type
;
on a
pseudospherical surface are normal to a Wsurface are respectively P\ + c
Pl
PO
the relations between the radii
=
.
a tanh
(cf.
,
a
7(3).
pl
p2
=
.,
a coth
Pi 4 c
,
a
where a and
10.
c are
constants
Show
that the linear elements of the second surfaces of center of the
are reducible to the respective forms
2
du"
&gt;Fsurfaces
of Ex.
ds.?
=
tanh 4 a
+
sech 2  dv 2
a
,
ds.?

coth 4
U
a
du 2
+
csch 2 ~ du 2
a
,
and that consequently these surfaces are applicable whose meridians are defined by
to surfaces
of
revolution
where
K
denotes a constant.
GENERAL EXAMPLES
11.
319
of the curves
Determine the particular form of the linear element (49), and the nature upon the surface to which the asymptotic lines on the sheets of
the evolute correspond,
(a)
when
p.
=
const;
(6)
11  = const.
Pi
Pz
Pz
a JFsurface is of the type (72), the surfaces of center are applicable to one another and to an imaginary paraboloid of revolution.
12.
When
13.
When
a IFsurface
is
has the form (VI,
GO), the curves
of the type (72) and the linear element of the sphere u+v const, and u const, on the spherical y
representation are geodesic parallels whose orthogonal trajectories correspond to the asymptotic lines on the surfaces of center hence on each sheet there is a family of geodesies such that the tangents at their points of meeting with an asymptotic
;
line are parallel to a plane,
which varies
in general
with the asymptotic
line.
14.
Show
that the equations
+ Jf Vs mdv. a a
y
= aUsina
C
J
where a denotes an arbitrary constant, define a family of applicable molding surfaces.
15.
When
is
lines of the
latter
the lines of curvature in one system on a surface are plane, and the second system lie on spheres which cut the surface orthogonally, the a surface of Joachimsthal.
16. The spherical lines of curvature on a surface of Joachimsthal have constant geodesic curvature, the radius of geodesic curvature being the radius of the sphere on which a curve lies. 17.
When
the lines of curvature in one system on a surface
is
lie
on concentric
its
spheres, it is a surface of Monge, whose directordevelopable vertex at the center of the spheres and conversely.
;
a cone with
18. The sheets of the evolute of a surface of Monge are the directordevelopable and a second surface of Monge, which has the same directordevelopable and whose
generating curve
is
the evolute of the generating curve of the given surface.
19. If the lines of curvature in one system on a surface are plane, the second system are plane, then all in the latter system are plane.
and two
in
20.
A
A
surface with plane lines of curvature in both systems, in one of which
circles, is
they are
(a)
(5)
surface of Joachimsthal.
locus of the orthogonal trajectories of a family of spheres, with centers
The
on a straight line, which pass through a circle on one of the spheres. (c) The envelope of a family of spheres whose centers lie on a plane curve C, and whose radii are proportional to the distances of these centers from a straight
line fixed in the plane of C.
21. If an arbitrary curve C be drawn in a plane, and the plane be made to move way that a fixed line of it envelop an arbitrary space curve T, and at the same time the plane be always normal to the principal normal to T, the curve C
in such a
describes a surface of Monge.
320
TFSUKFACES
S
are surfaces of
Enneper
23.
22. If all the Bianchi transforms of a pseudospherical surface 134), S is a surface of revolution. (cf. Ex. 5,
When
u has the value
in Ex. 5,
134, the surfaces
with the spherical
representation (25), and with the linear element
ds*
=
(HI cos w
+
Ydu 2 + U? sin 2 u du 2
,
where U\
is
an arbitrary function of
M, are surfaces of Joachimsthal.
same spherical representation
surface,
24. If the lines of curvature in both systems be plane for a surface S with the of its lines of curvature as for a pseudospherical
S
is
a molding surface.
25. If
its lines
S
is
of curvature,
a pseudospherical surface with the spherical representation (25) of and the curves v = const, are plane, the function 6, given by
c2w
sin 6 a
2
+
aw aw
cos
.
h sin
w
aw
dv
aw dv
= 0,
.
determines a transformation of Bianchi of S into a surface Si for which the lines
of curvature v
=
const, are plane.
26. necessary and sufficient condition that the lines of curvature v= const, on a pseudospherical surface with the representation (25) of its lines of curvature be spherical is that \r a.,
A
cot w
= y1 +
J
/
,
sin
w dv
that
where V and V\ are functions of and of i aw
sin 2 w aw a/
i
v alone.
a
Show
i
when w
is
a solution of (23)
aw\
aw\
aw \sinwatv aw\ a2 / i aM 2 \sinwau/
au\sin 2 wau/
the curves v
=
const, are plane or spherical,
and that
in the latter case
V and
V\
can be found directly.
27.
Show
that
when w
is
a solution of (23) and of
dv aucv 2
a
2
awau
du \au/
and
du \cos w dv/
(
)
^. 0,
the lines of curvature
u
=
const, are spherical
;
on the pseudois
spherical surface with the spherical representation (25) a function, upon the surfaces with the linear element
and that when w
such
/aw\ 2
or
/
r)w\ 2
)
\dv/
I
(sin
\
w
+F
du 2
+
cos w
f
V +V
former
dv/
where
is a function of alone, the curves case the spheres cut the surface orthogonally.
t&gt;
F
t&gt;
=
const, are spherical; in the
CHAPTER IX
DEFORMATION OF SURFACES
Problem of Minding. Surfaces of constant curvature. Ac 43 two surfaces are applicable when a onetoone cording to
135.
correspondence can be established between them which is of such a nature that in the neighborhood of corresponding points corresponding figures are congruent or symmetric. It was seen
that two surfaces with the same linear element are
applicable,
the parametric curves on the two surfaces being in correspon dence. But the fact that the linear elements of two surfaces are
unlike
is
in evidence of this
not a sufficient condition that they are not applicable we have merely to recall the effect of a change
;
by Minding
of parameters, to say nothing of a change of parametric lines. Hence we are brought to the following problem, first proposed *
:
To find a necessary and
applicable.
sufficient condition that
two surfaces be
From
the second theorem of
is
condition
follows that a necessary that the total curvature of the two surfaces at corre
it
64
sponding points be the same.
is
We
shall
show
all
that this condition
sufficient for surfaces of constant curvature.
In
64 we found that when
K
is
zero at
the surface
is applicable to the plane. the system of straight lines parallel to the rectangular axes,
points of a surface, If the plane be referred to
its
linear element
is
ds
797272 =dx +
2 2
dy*.
Hence the
surface
analytical problem of the application of a developable upon the plane reduces to the determination of orthogonal
systems of geodesies such that when these curves are parametric the linear element takes the above form.
*
Crelle, Vol.
XIX
(1839), pp. 371387.
321
322
DEFORMATION OF SURFACES
39,
Referring to the results of
factor
tt
v
we
see that in this case the
unity. Consequently we must find a function 6 such that the lefthand members of the equations
must equal
du
+ +
.dv
=
d(x
+ iy},
du
\
which case these equations give x and y by quadratures. Hence we must have
are exact differentials, in
du\
^E
to
99
which are equivalent
du~Hu
dv
ZEHdu F d_E_ dd _ J_ d_G_ _ ~
2
H
dv
^H ~du
2
EH dv
From
(V, 12)
K= 0.
seen that these equations are consistent when In this case 6, and consequently x and y, can be found by
it is
quadratures. The additive constants of integration are of such a character that if ar , y are a particular set of solutions, the most general are
x
=x
cos a
yQ
sin a
+
a,
y
=X
Q
sin
a
+y
Q
cos a
+
/&gt;,
where
of
a, #, 5 are
arbitrary constants.
effect the isometric representation
In the above manner we can
any developable surface upon the plane, and consequently upon itself or any other developable. These results may be stated thus
:
A developable surface is
applicable
to itself,
or to
any other develop
able, in a triple infinity of ways,
and
the complete determination of
the applicability requires quadratures only.
Incidentally
we have
the two theorems:
be foundby quadratures.
The geodesies upon a developable surface can
If the
total
curvature of a quadratic
form
be zero, the quadratic
form
is
reducible by quadratures to dad&.
SUEFACES OF CONSTANT CURVATURE
323
Suppose now that the total curvature of two surfaces S, Sl is 2 I/a where a is a real constant. Let P and 7? be points on S and /S^ respectively, C and C geodesies through these respective and C and C for the points, and take P and I[ for the poles of a polar geodesic system on these surfaces. The curves v =
,
l
l
linear elements are accordingly (VIII, 6)
d**
= du + sin
2
2
 dv\
u
ds 2
v
=
du* + sin 2
v
^ dv
2
.
Hence
the equations
u
=
to
determine an isometric representation of one surface upon the
other, in
which
P
and C correspond
P
in
and Cl respectively.
the second equation
According as the upper or lower sign
is
used, corresponding figures are equal or symmetric. Similar results obtain for pseudospherical surfaces. Hence we have:
Any two surfaces of constant curvature, different from zero, are in two ways applicable so that a given point and geodesic through it on one surface correspond to a given point and geodesic through it on the other.
itself so that a
In particular, a surface of constant curvature can be applied to given point shall go into any other point and a
geodesic through the former into one through the latter. Combin 117, we have: ing these results with the last theorem of
nondevelopable surface of constant curvature can be applied to in a triple infinity of itself, or to any surface of the same curvature,
A
ways, and the complete realization of the applicability requires the solution of a Iliccati equation.
136. Solution of the problem of Minding. proceed to the determination of a necessary and sufficient condition that two sur faces S, 8 of variable curvature be applicable. Let their linear
We
elements be
ds
2
= E du + 2 Fdudv + G dv
2
2
,
ds
2
=E
if
du
2
+ 2F
du dv
+G
dv 2
.
By
(1)
definition
S and S
are applicable
there exist two independ
ent equations
(/&gt;
(U, V)
=
$(U
J
,
V
1
),
^ (U,
V)
=
^&lt;(U
,
V
),
establishing a onetoone correspondence between the surfaces of such a nature that by means of (1) either of the above quadratic
forms can be transformed into the other.
324
DEF011MATION OF SURFACES
two surfaces are applicable, the differen formed with respect to the two linear elements are parameters
It is evident that if the
tial
equal.
(2)
Hence
a necessary condition
A;&lt;/&gt;
is
A^ =
,
A^,
f)=A!(&lt;#&gt; ,
f),
A f=A;t
1
.
,
where the primes indicate functions pertaining to S
These con
ditions are likewise sufficient that the transformation (1) change either of the above quadratic forms into the other. For, if the
curves
(/&gt;
= const.,
^r
const.
;
&lt;/&gt;
= const., ^ = const,
f
be taken for
the parametric curves on S and S linear elements may be written (cf.
,
respectively, the respective
37)
Hence when equations
The next step is Since the curvature of two applicable surfaces at corresponding points is the same, one such equation is afforded by the necessary condition
(3)
the surfaces are applicable. the determination of equations of the form (1).
(1) (2) hold,
and
K(u,v)
first
= K (u ,v
.
).
The
(4)
of equations (2)
is
A^A;*
K
37),
Both members
this case the curves
(
of this equation cannot vanish identically. const, would be const, and
For, in
K=
minimal
are
and consequently imaginary.
is,
If these
two equations
independent of one another, that
they establish a correspondence, and the condition that metric is, as seen from (2),
it
be
iso
If,
however,
(5)
\K
for the second of (1)
we may take
(6)
unless
(7)
A
2J
fiT
PROBLEM OF MINDING
If this condition
325
(3), (6)
be not satisfied, the conditions that
define
an isometric correspondence are
AA
1
a
JST
=
Finally, we consider the case where both (5) and (7) hold. Since the ratio of and A 2 is a function of JC, the curves const.
\K
K
K=
form an isothermal sys trajectories the function t can be found 41). Moreover, by quadratures, and the linear element is reducible to
t
and their orthogonal tem of lines on S (
= const,
(8)
ds
2
=2
2
J( K
(dK
)
+e
J
&lt;
A
&gt;
dt
2
).
When
in particular
A ^T=
0,
the linear element
is
In like manner the linear element of
S
is
reducible to
or, in
the particular case
A^
= 0,
to
In either case the equations
K=K
where a
surfaces.
is
i
t
=
t
+a,
an arbitrary constant, define the applicability of the
all
We
have thus treated
possible cases and found that
it
can
be determined without quadrature whether two surfaces are appli cable. Moreover, in the first two cases the equations defining the correspondence follow directly, but in the last case the determina
tion requires a quadrature. The last case differs also in this respect the application can be effected in an infinity of ways, whereas in
:
the
first
two cases
it is
unique.
&lt;r
* If the surface be referred to the curves
where a
that
Vol.
AI(&lt;T,
J ~vf(K)
&&lt;&lt;?)
C=
=
= const,
and
their orthogonal trajectories,
,
equation
,
A%&lt;r
(6)
may
be replaced by
A2
tr
= A^
,
and
it
can be shown
Cf.
Ai(&lt;r
)
is
a consequence of the other conditions.
Darboux,
Ill, p. 227.
326
DEFORMATION OF SURFACES
(8)
Furthermore, we notice from
face
that in the third case the sur
S
is
applicable to a surface of revolution, the parallels of the
latter corresponding to the curves
K= const,
of the former.
Con
versely, the linear element of every surface applicable to a surface of revolution can be put in the form (8). For, a necessary and sufficient condition that a surface be applicable to a surface of
revolution
is
that
its
linear element be reducible to
where
U is
a function of u alone
(
46).
Now
Cdu
A lW = l,
From
the
X= ,
U"
second
it
follows that
u
F(K), and consequently
F^K).
equations,
(9)
When
these values are substituted in the above
we
have, in consequence of Ex. 5, p. 91,
A
1 /C=/(A"),
A,JiT
=*&lt;*).
Hence we have the theorem
Equations
(9) constitute
to
:
surface be applicable
a necessary and sufficient condition that a a surface of revolution.
The equations
K=K,
Therefore
t
=
t
+a
:
define an isometric representation of a surface with the linear ele
ment
(8)
upon
itself.
we have
K = const,
is
Every surface applicable to a surface of revolution admits of a continuous deformation into itself in such a way that each curve
slides over itself.
itself in
Conversely, every surface applicable to
an infinity of
applicable to a surface of revolution. For, if the curvature ways is constant, the surface is applicable to a surface of revolution
135), and the only case in which two surfaces of variable curva ture are applicable in an infinity of ways is that for which condi tions (5) and (7) are satisfied.
(
DEFORMATION OF MINIMAL SURFACES
137. Deformation of
327
means
of determining the
minimal surfaces. These results suggest a minimal surfaces* applicable to a surface
of revolution. In the first place we inquire under what conditions two minimal surfaces are applicable. The latter problem reduces to the determination of two pairs of parameters, w, v and u^ v v and and Ffa^, ^ 1 (f ), which satisfy two pairs of functions, F(u),
&gt;
&lt;f&gt;(v)*
1
the condition
(10)
(1
+ uvfF(u)(v) dudv = (1 + u^)* f\(uj 4^) du^dv r
it
From
the nature of this equation
serve to establish the correspondence between the either of the form
(11)
follows that the equations which two surfaces are
^=0(M),
,= *(),
u and
l
V 1= ^(V),
or
(12)
,
= *()
and
if
If either set of values for
v l be substituted in (10),
removing the common factor dudv we take the logarithmic derivative with respect to u and v, we obtain
after
(1
+ u^Y
du dv l
l
(1
+ uvf
dudv
2
As
,..
this
may
be written
~~
o
(i + uft)*
(I +^^y
the spherical images of corresponding parts on the two surfaces are equal or symmetric according as (11) or (12) obtains ( 47). The
latter case reduces to the
former when the sense of the normal to
either surface
is
changed.
When
this has
been done, corresponding
spherical images are equal and can be made to coincide by a rota tion of the unit sphere about a diameter. Hence one surface can be
so displaced in space that corresponding normals become parallel, in which case the two surfaces have the same representation, that
is,
Wj
= u,
is
vx
= v. Now
equation (10)
is
which
equivalent to
F (u)=cF(u),
l
3
*
no.
328
DEFORMATION OF SURFACES
i&lt;x
where c denotes a constant. If the surfaces are real, c must be of the form e Hence, in consequence of 113, we have the theorem
.
:
minimal surface admits of a continuous deformation into an which are either associate to it or can infinity of minimal surfaces,
be
A
made such by a
suitable displacement.
pass to the determination a continuous deformation into of
We
of a
minimal surface which admits
and consequently is appli In consequence of the interpre cable to a surface of revolution. tation of equation (13) it follows that if a minimal surface be deformed continuously into itself, a point p on the sphere tends to
itself,
move
in the direction of the small circle
the momentary small circles moves over
axis of rotation,
itself.
through p, whose axis is and consequently each of these
47
it
From
follows that
if
the
axis of rotation be taken for the 2axis, these small circles are the curves uv = const. In the deformation each point of the surface
moves along the curve
tion of uv.
through From (VII, 100, 102) we have
K= const,
it.
Hence
K
is
a func
A= ___^l_
uF
The common
denoted by
K,
;
function of uv, and hence consequently F(u)Q(v) must be a
(u) __
v&lt;&
(v)
F(u)
value of these
4&gt;(v)
two terms
is
a constant.
K
,
If it
be
we have
where
c
and
c1
are constants.
Hence from (VII,
to
98)
we have
:
Any minimal
surface applicable
a surface of revolution can be
the defined by equations of
form
&lt;?j
*
c
f (1 _ u *) u du + 
f (1  v
1 )
v*dv,
(14)
y
c Cfl + u^u du
2
/r
J
{
^Ci 1(1
&lt;+i
2V
iv,
u ^du
+
c^Ji
constants.
ivhere
c,
c.,
and K are arbitrary
DEFOKMATION OF MINIMAL SUEFACES
Since the curves
329
on the sphere by the the zaxis, in each finite deformation of the surface into itself, as well as in a very small one, the unit sphere undergoes a rotation about this axis. In 47 it was seen that such ia ia a rotation is equivalent to ve~ where a replacing u, v by ue denotes the angle of rotation. Hence the continuous deformation
const, are represented
is
K
small circles whose axis
,
,
of a surface (14) is defined by the equations resulting from the substitution in (14) of ue ia ve~ for u, v respectively. An important property of the surfaces (14) is discovered when
i&lt;z
,
zaxis.
its
we submit such a surface to a Let S denote the surface
equations in the
rotation of angle a about the
in its
new
position,
and write
form
f (1  tf
and similarly
for
y and
z.
Between the parameters
ta
,
u, v
and
u, v
the following relations hold:
u
and we have
x
also
= ue
v
= ve~
tar
,
= x cos a
y
sin a,
~y
= x sin a f y cos
(14),
5&gt;
a:,
z
= z.
Combining these equations with
we
(v)
find
F (u) =
is
cu e~
K
ia(K
+
2)
,
=
K c { v e ia(K +
2)
.
Hence, for the correspondence defined by u = ?/, v = v, the surface S an associate of S, unless K f 2 = 0, in which case it is the same
surface.
We
are
(cf.
consider the latter case, and remark that
its
equations
110)
be replaced by ue ia ve~ ia and the resulting expressions be denoted by x v y^ z x we have
If u, v
, ,
,
(15)
x l = xcosa
ysina
1
y^
= x sin a + y cos a,
zl
= z + lR(iac}.
c
Hence, in a continuous deformation, the surface slides over itself with a helicoidal motion. Consequently it is a helicoid. Moreover, it is the minimal helicoid. For, every helicoid is applicable only
330
DEFORMATION OF SURFACES
and each minimal surface applicable
to
to a surface of revolution,
a surface of revolution with the zaxis for the axis of revolution
2 will the of the sphere is defined by (14). But only when ic = ia a set of equations such as (15). ve~ substitution of we give
1
",
Hence we have
:
The helicoidal minimal surfaces are defined by the Weierstrass
formulas when F(u)=c/u
2
.
And we may
state the other results thus
:
applicable to a surface of revolution, be rotated through any angle about the axis of const. the unit sphere whose small circles represent the curves
If any nonhelicoidal minimal surface, which
is
K
on the surface, and a correspondence with parallelism of tangent are associate ; con planes be established between the surfaces, they a minimal surface are supcrposable. sequently the associates of such
EXAMPLES
1
.
Find under what conditions the surfaces, whose equations are
z\
F(r)
+
av,
can be brought into a onetoone correspondence, so that the total curvature at surfaces corresponding points is the same. Determine under what condition the
are applicable.
tangent planes to two applicable surfaces at corresponding points are the surfaces are associate minimal surfaces. parallel,
2.
If the
3.
Show
that the equations
x
=
ea u t
y
=
e a v,
z
aea u z
f
b&lt;y~
av2
,
where a is a real parameter, and a and 6 are constants, define a family of parab oloids which have the same total curvature at points with the same curvilinear
coordinates.
4.
Are
these surfaces applicable to one another
Find the geodesies on a surface with the linear element d u z _ 4 y dudv + 4 u dv 2 2
ds =.
2
4(wt&gt; )
6 in
?
Show
5
that the surface
is
the applicable to a surface of revolution, and determine
latter.
form of a meridian of the
.
Determine the values of the constants a and
ds*
=
du 2
+
[(u
+
au)
2
+
6 2 ] du 2 ,
so that a surface with this linear element shall be applicable to
(a)
(&)
the right helicoid. the ellipsoid of revolution.
SECOND GENERAL PROBLEM
6.
331
and sufficient condition that a surface be applicable to a surface that each curve of a family of geodesic parallels have constant geodesic curvature.
of revolution
is
A necessary
7.
Show
that the helicoidal minimal surfaces are applicable to the catenoid and
to the right helicoid.
have seen that 138. Second general problem of deformation. can always be determined whether or not two given surfaces are applicable to one another. The solution of this problem was an
it
We
important contribution to the theory of deformation. An equally important problem, but a more difficult one, is the following
:
To determine
all the surfaces applicable to
a given one.
This problem was proposed by the French Academy in 1859, and has been studied by the most distinguished geometers ever since. Although it has not been solved in the general case, its profound study
has led to
If
many
interesting results,
some
of
which we
shall derive.
the linear element of the given surface be
2
c?s
= Edu? + 2 F dudv
it is
f
G dv*,
this
every surface applicable to
determined by
2
,
form and by a
dudv + D"dv whose coefficients satisfy second, namely Ddu*+ 2 the Gauss and Codazzi equations ( 64). Conversely, every set of of these equations defines a surface applicable solutions D, ,
D
D
D"
to the given one, and the determination of the Cartesian coordinates of the corresponding surface requires the solution of a Riccati equa
But neither the Codazzi equations, nor a Riccati equation, can be integrated in the general case with our present knowledge of differential equations. Later we shall make use of this method in
tion.
the study of particular cases, but for the present we proceed to the exposition of another means of attacking the general problem. n obtained from the Gauss equations When the values of D,
DD
,
(V, 7) are substituted in the equation
H K=J)D"D
2
2
,
the result
2
ing equation is reducible, inconsequence of the identity (cf. Ex. 6, p. 120), to
(16)
A^ =1 A
dtf
"
\du
dv~\
1J
^
I 2 J dv
332
DEFORMATION OF SURFACES
is
This equation, which
E, F,
satisfied also
by y and
2,
involves only
its
G and
their derivatives,
and consequently
,
integration
complete solution of the problem. It is linear in 2 / c x Yl &x tfx tfx , , ^, \tfxtfx , , ^ri^l^rsr) J dir 1 TT Tl and therefore is of the form oTi cv 0tr \dudv / \_du studied by Ampere. Hence we have the theorem:
will give the
^&lt;7v
the integration of
The determination of all surfaces applicable to a given one requires a partial differential equation of the second order
of the
Ampere
type.
In consequence of (16) and (V, 36) of a surface with the linear element
(17)
we have
that the coordinates
ds
2
= Edu + 2 Fdudv + G dv
2
2
are integrals of
(18)
A
22
= (l
1
A^JST,
the differential parameters being formed with respect to (17). shall find that when one of these coordinates is known the other
We
two can be found by quadratures.
Our general problem may be
Given three functions E, F,
x, y,
stated thus
:
G
of
u and
v ; to find all functions
z of
u and
dx
2
v
which
1
satisfy the equation
2
+ dy + dz = E du* + 2 Fdudv + G dv\
may
be chosen arbitrarily.
where du and dv
Darboux
(19)
* observed that as the equation
may
2
be written
dx
2
+ dy* = Edu +2Fdudv + Gdv  dz\
2
whose lefthand member
is
the linear element of the plane, or of a
developable surface, the total curvature of the quadratic
(20 )
form
\E p?Y"U + 2\F dudv\ dudv **\ L dv\ L
WJ
(
*
must be zero
known,
64).
this,
In order to find the condition for
we assume
that z
is
and take for parametric lines the curves z = const, and
*L.c., Vol. Ill, p. 253.
their
SECOND GENERAL PROBLEM
orthogonal trajectories for
ters the righthand
v
333
= const. With
this choice of
2
parame
The
of (19) reduces to (E I)dz condition that the curvature of this form be zero is
member
+ Gdv*.
tr
where
K
denotes the curvature of the surface.
But
2
.
this is the
condition also that z be a solution of (18)
when
the differential
parameters are formed with respect to
Edz 2 }Gdv
However, the
;
members
z is
of equation (18) are differential parameters consequently a solution of this equation whatever be the parametric curves.
By
reversing the above steps
z is
we prove
the theorem
:
When
any
integral of the equation (18), the quadratic form (20)
has zero curvature.
is known we can find by quadratures two functions x, y such that the quadratic form (20) is (cf. 135) z equal to dx* + dy provided that
When
such a solution
,
that
is,
Ajg
&lt;
1.
Hence we have the theorem
:
If
z be
a solution of
A 22
(1
A
X
0)
K such that A^
&lt;
1, it is
one
a surface with the given linear ele the other two coordinates can be obtained by quadratures. ment, and
of the rectangular coordinates of
139. Deformations
which change a curve on the surface
into a
given curve in space.
We
consider the problem
:
Can a surface be deformed in such a manner that a given curve C upon it comes into coincidence with a given curve F in
Let the surface be referred to a family of curves orthogonal to C and to their orthogonal trajectories, C being the curve v = 0, and
its arc
being the parameter u, so that conditions hold for F on the deform.
E
\ for v
= 0. The
same
334
DEFORMATION OF SURFACES
(
Since the geodesic curvature of C is unaltered in the deformation for the new surface, 58), it follows from the equation (IV, 47)
namely
(21)
p
is
= p g amw,
if
that the deformation
impossible,
the curvature of
F
at
any
point
point.
to,
is
than the geodesic curvature of C at the corresponding Since both p and p g are known, equation (21) determines
less
and consequently the direction of the normal
to the
new
surface
along
F
is fixed.
to the curves
u
=
This being the case, the direction of the tangents const, on the new surface at points of F can be
.
found, and so
as well as
we have
&gt;
1
V&lt;?
dx

1
c&gt;f
1
the values of
to
ri
=
cz
\IG
co
vV;
w
tor v
= U,
for v
= 0,
cu
cu
du
the latter being the directioncosines
of the tangent to F.
respect to u,
for
we
obtain the values ot
If these expressions , tfx c~y
^
3u"
cu"
be differentiated with 2 2 2 cz cx dz tfy T r 7777, cudv ducv cucv
;
&gt;
cu"
v=0.
Since
F=Q
are
tfx
and
E=1
SEdx
to to
for v
= 0,
the Gauss equations
(V, 7) for v
=
=
J.
"
1
du
2
2G
I/
r JJJL+
Jjy
t/*f
J
*y**
f**
2
"a^
a^
2G
2
du to
dv*
2 cu du
6r
00 dv
All the terms of the first two equations have been determined hence the latter are given by these equations. except D and D
1 ;
Since the total curvature
known
/&gt;/&gt;"
/&gt;
and p =p
unaltered by the deformation, it is = at all points of F; consequently // is given by H*K 2 is zero, in which case F is an asymptotic line , unless 2 dx from is found we can obtain the value of When 5
A"
is
D
.
/)"
(J
the last of equations (22). From the method of derivation of equa tion (16) it follows that the above process is equivalent to finding
the value of
^ from
to
equation (16), which
is
possible unless
D=
0.
Excluding
this exceptional case,
we remark
that
if
equations (22)
PARTICULAR DEFORMATIONS
be differentiated with respect to u,
335
all
we
obtain the values of
the
derivatives of x of the third order for v
=
except

The
latter
be obtained from the equation which results from the differ entiation of equation (16) with respect to v. By continuing this of the derivatives of x of all process we obtain the values for v
may
orders,
and likewise of y and
z.
If
we
,
indicate
the values of functions,
when u = UQ
n 2\#ir/o
v
= 0,
by subscript null the expansions
= z + fdx\
\dufy
idx
\dvfy
\dudvh
as of
:
and similar expansions for y and 2, are convergent in general, Cauchy has shown,* and x, y, z thus defined are the solutions
equation (16)
which for
can
be
v
=
satisfy the given conditions.
Hence
A
it
surface
S
deformed in such a manner that a curve C upon
curvature of
of
comes into coincidence with a given curve F, provided that the F at each point is greater than the geodesic curvature
C
at the corresponding point.
ormation
There remains the exceptional case p is possible, F is an asymptotic
=p
.
a
If the desired def
line
on the deform, and
is
consequently, by Enneper s theorem ( must satisfy the condition r 2 = 1/JC.
is
59), its radius of torsion
Hence when C
D"
given,
F
determined,
If
F
to be an asymptotic line. satisfies these conditions, the value of
if it is
for v
=
is arbi
trary, as
we have
seen.
But when
it
it
has been chosen, the further
determination of the values of the derivatives of
order for v
=
is
the general case. of these surfaces, depending upon an arbitrary function. For all .of these surfaces the directions of the tangent planes at each
#, y, z of higher the same as that pursued in unique, being Hence equation (16) admits as solution a family
point of
F
are the same.
Hence we have the theorem
:
Criven a curve
curve
tion of
F with 8 in an
to
C upon a surface 8 ; there exists in space a unique which C can be brought into coincidence by a deforma
infinity of
ways ; moreover,
all the
new surfaces are
tangent
.
one another along F.
aux derivees partielles du second
ordre, chap.
* Cf Goursat, Lemons surT integration des Equations ii. Paris, 1896.
336
If
DEFOKMATION OF SURFACES
C
is
an asymptotic line on
S, it
may
be taken for F; hence
:
A
surface
may
be
subjected
to
which a given asymptotic line is be an asymptotic line on each deform.
a continuous deformation during unaltered in form and continues to
This result suggests the problem
:
Can a surface be subjected to a continuous deformation in which a curve other than an asymptotic line is unaltered?
By hypothesis the curvature is not changed and the geodesic curvature is necessarily invariant; hence from (21) we have that
sin
o&gt;
must have the same value
for all surfaces,
for all the surfaces.
If
o&gt;
is
the
the tangent plane is the same, and consequently same the expansions (23) are the same. Hence all the surfaces coincide in this case. However, there are always two values of W for which
sin
o&gt;
has the same value, unless
w
is
a right angle.
Hence
it is
have two applicable surfaces passing through a curve whose points are selfcorrespondent, but not an infinity of such
possible to
surfaces.
Therefore
:
An asymptotic line is the only curve on a surface which can remain unaltered in a continuous deformation.
140. Lines of curvature in correspondence.
We
C upon
in such a manner become a line of curvature on the new surface. may Suppose it is possible, and let F denote this line of curvature. The radii of curvature and torsion of F must satisfy (21) and dto/ds 1/T (cf. 59), where p g is the same for F as for C. If we choose for w any function whatever, the functions p and r are
a surface
S can be deformed
inquire whether that a given curve
it
;
thus determined, and
F
is
unique.
Since
o&gt;
fixes the direction of the
tangent plane to the new surface along F, there is only one deform of S of the kind desired for each choice of w (cf. 139). Hence
:
surface can be deformed in an infinity of ways so that a given curve upon it becomes a line of curvature on the deform.
A
This result suggests the following problem of Bonnet*: To determine the surfaces which can be deformed with preservation
of their lines of curvature.
I
* Memoire sur la theorie des surfaces de applicables sur une surface donnee, Journal Ecole Poly technique, Cahier 42 (1867), p. 58.
LINES OF CUBVATUBE IN COBBESPONDENCE
337
We
follow the
method
of
Bonnet
in
making use
of the
funda
mental equations in the form (V, 48, 55). We assume that the lines In this case these equations reduce to of curvature are parametric.
($&gt;i
_
dv
dc
t~~
Sr
^
(24)
From these equations it follows that if $ and S are two applicable surfaces referred to corresponding lines of curvature, the functions r and r l have the same value for both surfaces, and consequently
the same
is
true of the product
qp r Hence our problem reduces
;
to the determination of
the above equations.
(25)
p v q p[, q In consequence of the identity
p(q
two
sets of functions
,
satisfying
=P&
we have from
the
first
two
of (24)
*&lt;f&gt;(u)
&lt;S?S&lt;f+
&lt;#&gt;(M),
2 2 where f(v) of which the integrals are p i = p?+f(v), q = and are functions of v and u respectively. The parameters w, v may be chosen so that these functions become constants #, /3,
and consequently
2
(27)
If these equations
;&gt;I
= K+
2
2
=&lt;Z
?
+is
be multiplied together, the resulting equation reducible by means of (25) to either of the forms
(28)
pi ft
+ (fa + a(3 = 0,
p(*P
+ z *aa& = b.
the first we see that a and fi cannot both be positive if S is real, and from the second that they cannot both be negative. We assume that a is negative and j3 positive, and without loss of generality write
From
(29)
rfrfli ^tf+lq*
The first of (28) reduces to pi we introduce a function thus
o&gt;,
= l.
In conformity with this
p = cosh
l
&&gt;,
q
= sinh
o&gt;.
Then
equations (29)
may
be replaced by
ft),
= sinh p[
^
= cosh
o&gt;.
338
DEFORMATION OF SURFACES
Moreover, the fundamental equations (24) reduce to
_
dv
a
2
o&gt;
du
V]
,
o&gt;
du
,
:
a
H
c  =
o&gt;
o2
smh
.
,
cosh
o&gt;.
dt*
0v
Comparing these
results with
118,
we
see that the spherical
f
representation of lines of curvature of the surfaces S and S respec tively is the same as of the lines of curvature of a spherical surface
that every surface of this kind admits of an applicable surface with lines of
its
and of
Hazzidakis transform.
Conversely,
we have
curvature in correspondence.
The preceding
the
first
investigation rested on the hypothesis that neither
nor second of equations (24) vanishes identically. Suppose that the second vanishes then q is a function of u alone, say $(u). Since the product p^q differs from the total curvature only by a
;
factor
(cf.
70),
is
Equation (25) nated from this equation and the
also
is
p cannot be zero now of the form p
l
;
therefore r
l &lt;f&gt;(u)=p[&lt;t&gt; l
=
and q
^(u).
(u).
If p[ be elimi
first of (27), it is
found that
p
l
a function of
u
alone.
Hence the curves
v = const,
on the
sphere are great circles with a
is
a molding surface
that
find
we may
take q
h #,
common diameter, and therefore S The parameter u may be chosen so 130). ( = 1 and p^=U\ then from (27) and (25) we
where a
is
= Vf/
/&gt;(
2
q
= U/^/U* + a,
:
an arbitrary constant.
Hence we have the theorem
necessary and sufficient condition that a surface admit of an is that applicable surface with lines of curvature in correspondence its lines of cur the surface have the same spherical representation of
vature as a spherical surface 2, or be a molding surface ; in the first case there is one applicable surface, and the spherical representation
A
of
its lines
of curvature
is
the
of
2
;
in the second case there
is
same as of the Hazzidakis transform an infinity of applicable surfaces.*
141. Conjugate systems in correspondence.
When two
surfaces
are applicable to one another, there is a system of corresponding lines which is conjugate for both surfaces (cf. 56). The results
of
140 show that
for a given conjugate system
* Cf EX. 14, p. 319.
.
on a surface S
CONJUGATE SYSTEMS IN COEEESPONDENCE
there
is
339
not in general a surface
S
l
applicable to
S with
the corre
sponding system conjugate. We inquire under what conditions a given conjugate system of S possesses this property. Let S be referred to the given conjugate system. If the corre sponding system on an applicable surface Sl is conjugate, we have
D = D[= 0,
for the total curvature of the
this equation
Dip? = DD"
is
;
two surfaces
the same.
We replace
by the two
D!  tanh 6 D,
D[
= coth 6
.
D",
thus defining a function
6.
The Codazzi equations
for
S
are
Since these equations must be satisfied by
30
D
l
and
Z
D",
we have
22
lz
The condition
D
,
c6
fll
of integrability of (30)
is
reducible to
2
D
m
As
to
the two roots of this equation differ only in sign, and thus lead symmetric surfaces, we need consider only one. If it be substi
tuted in (30),
we
are necessary in order that
obtain two conditions upon E, F, G 7), which S admit of an applicable surface of the
;
D",
Hence in general there is no solution of the problem. the two expressions in the brackets of (31) vanish identically, the conditions of integrability of equations (30) are completely satisfied, and S admits of an infinity of applicable sur
kind sought.
However,
if
faces
upon which the coordinate curves form a conjugate system.
:
Consequently we have the theorem
If a conjugate system on a surface S corresponds to a conjugate system on more than one surface applicable to S, it corresponds to a
conjugate system on an infinity of surfaces applicable
to S.
340
DEFORMATION OF SURFACES
shall give this result another interpretation by considering the spherical representation of S. From (VI, 38) we have
We
ii/^ = la/
{rtV
ment
&gt;
1221
.D
f!2V
are
firiD".
/12V
la/3"
~li/
ele
formed with respect to the linear
If
of the spherical representation of S. values in (30), we get
we
substitute these
d6
= sf!2\ f
12J
tanh
d6
0,
... /12V coth 0, =\ \
a^
0*
11
J
and the condition that these equations have an integral involving a parameter becomes
a
ri2V_
a
ri2V
Sii/~Sla./
The
py ri2V
"iJisr
first of these equations is the condition that the curves the sphere represent the asymptotic lines upon a certain sur upon denotes the total curvature of face 2 (cf. Moreover, if 78).
K
S, and we put
(34)
K=
2
l//&gt;
we have
2
du
Now
equations (33) are equivalent to (34), and
& log p
r,&lt;2
which reduces
is
to
=
cucv
),
0.
As
and
the general integral of this equation
p
= cf)(u) 4 ^(i
v respectively,
we
are arbitrary functions of u and have the following theorem due to Bianchi*
where
(/&gt;
^
:
necessary and sufficient condition that a surface S admit a con tinuous deformation in which a conjugate system remains conjugate
is
A
totic lines
that the spherical representation of this system be that of the asymp of a surface whose total curvature, expressed in terms of
to
parameters referring
these lines, is of the
form
*
Annali, Ser.
2,
Vol.
XVIII
(1890), p. 320; also Lezioni, Vol. II, p. 83.
CONJUGATE SYSTEMS IN COKKESPONDENCE
The pseudospherical
341
K of
this form.
In this case
("12")
surfaces afford an example of surfaces with and ^r are constants, so that equa&lt;/&gt;
tions (34) reduce to
f 12"
1
\
~\
a
**
&gt;

^ which, in consequence of
f {11"
= 1f22
^,
tions that the parametric curves a conjugate system of geodesies state these results thus
:
f=0. But these are the condi on S be geodesies. A surface with
is

called a surface of
/r
l
oss.
We
of l^oss admits of a continuous deformation in which the geodesic conjugate system is preserved ; consequently all the new sur faces are of the same kind.
A surface
EXAMPLES
1.
Show
that every integral of the equation Ai0
(18).
=
1 is
an integral of the funda
mental equation
2. On a right helicoid the helices are asymptotic lines. Find the surfaces appli cable to the helicoid in such a way that one of the helices is unaltered in form and
continues to be an asymptotic
3.
line.
A
on the two surfaces
4.
surface applicable to a surface of revolution with the lines of curvature in correspondence is a surface of revolution.
that the equations
Show
X=KTCOS,
K
y
=
train,
K
z
= /Vl J
/c
2r 2
c?M,
define a family of applicable surfaces of revolution with lines of curvature in corre spondence. Discuss the effect of a variation of the parameter K.
5.
applicable to
Let S denote a surface parallel to a spherical surface S with preservation of the lines of curvature.
S.
Find the surface
6. It Si and S2 be applicable surfaces referred to the common conjugate sys tem, their coordinates &i, y\,z\\ 2 ?/ 2 2 are solutions of the same point equation (cf. VI, 26), and the function xf f y? + zf (x + y. + z) also is a solution.
,
,
7.
Show
corresponding points on the surfaces Si and parametric lines form a conjugate system.
applicable to Si and
8.
/S 2
that the locus of a point which divides in constant ratio the join of 2 of Ex. 6 is a surface upon which the
&lt;S
Under what condition
is
this surface
?
The tetrahedral surface
x
= A(a + u)*(a + )*, y admits of an infinity of deforms
The curves u
=
B(b + u)*(b +
v)*,
z
=
C(c
+
w)*(c
+
f
v)
,
=
v
upon these surfaces are congruent, and consequently each
it.
is
an
asymptotic line on the surface through
342
9.
DEFORMATION OF SURFACES
If the equations of a surface are of the
form
x
the equations
= U1 Vll
y
= UiV!,
z=V*
t
sin 0,
where h denotes a constant, define a family of applicable surfaces upon which the parametric lines form a conjugate system.
10.
Show
that the equations of the quadrics can be put in the form of Ex.
results to this case.
9,
and apply the
142. Asymptotic lines in correspondence.
Deformation of a ruled
139) that a surface can be subjected to a continuous deformation in which an asymptotic line remains
surface.
(
We
have seen
ask whether two surfaces are applicable with asymptotic. the asymptotic lines in one system corresponding to asymptotic
lines of the other.
We
We
assume that there are two such surfaces,
S,
S and we
lt
v
=
take the corresponding asymptotic lines for the curves const, and their orthogonal trajectories for u = const. In con
sequence of this choice and the fact that the total curvature of the two surfaces is the same, we have
(36)
J9
= D =0,
1
JF=0,
)
D = D[.
to
The Codazzi equations
&lt;
(V, 13
for
S reduce
Q
Because of (36) the Codazzi equation for S1 analogous to the first of (37) will differ from the latter only in the last term. Hence we
= must have either or Ef(u}. In the former case the sur faces S and Sl are congruent. Hence we are brought to the second, which is the condition that the curves v = const, be geodesies. As the latter are asymptotic lines also, they are straight, and conse quently 8 must be a ruled surface. By changing the parameter w, we have J5? = l, and equations (37) reduce to
D"
Z&gt;",
EULED SURFACES
By a suitable choice of the parameter v may be replaced by JX=1/V5, and the
the
first
343
of these equations
second becomes
= ra/i ~(I
J 9*\&
These results establish the
fol
:
is an arbitrary function. theorem of Bonnet lowing
where
&lt;/&gt;
sufficient condition that a surface admit an with the asymptotic lines in one system on each applicable surface surface corresponding is that the surface be ruled; moreover, a
A
necessary
and
ruled surface admits of a
generators remain
straight.
continuous deformation in which the
To
this
may
be added the theorem
:
the asymptotic lines in both on each surface are in correspondence, the surfaces are con systems
If two surfaces are applicable and
gruent, or symmetric.
This
is
readily proved
when
the asymptotic lines are taken as
parametric.
We
shall establish the
second part of the above theorem in
another manner.
ruled surface in
(38)
For this purpose we take the equations of the the form ( 103)
Q
x
,
=x
its
+lu,
and
y
= y +mu,
Q
z
= z +nu,
functions of
C expressed as m, n are the directioncosines of the generators, also functions of v. They satisfy the conditions
where XQ
y^, z
are the coordinates of the directrix
arc
v,
I,
(39)
aJ
1 +jtfK !.
2
*
+w +n = l,
a
a
where the accents indicate differentiation with respect to Furthermore, the linear element is
(40)
v.
ds*= du 2 + 2 cos

dudv n
+ (aV+
b
2
2
=
l
x
Hence
if
we have
problem
of finding a ruled surface applicable to
a ruled surface with the linear element (40), the it, with the gener
ators of the
nation of six functions of
two surfaces corresponding, reduces to the determi v, namely X Q y z I, m, n, satisfying
, ,
;
344
DEFORMATION OF SURFACES
the five conditions (39), (41). From this it follows that there is an arbitrary function of v involved in the problem, and consequently
there
is
an infinity of ruled surfaces with the linear element
(40).
There are two general ways in which the choice of this arbi either as determining the form of trary function may be made,
the directorcone of the required surface, or by a property of the consider these two cases. directrix.
We
143.
Method
of Minding.
/,
The
first
case
was studied by Mind
n
ing.*
(42)
He
/
took
m, n
cos
in the
i/r,
form
&lt;
=
cos
&lt;
m = cos
sin
i/r,
= sin
&lt;/&gt;,
which evidently satisfy the second of
reduces to
2
(39).
The
first
of
(41)
(43)
If
&lt;J&gt;
+^
2
cos
2
=a
a
.
solve equations (39) and (41) for x^ y Q z expressions are reducible by means of (VII, 63) to
,
we
,
the resulting
(44)
Q
=
I
cos 6 Q
+^
[I
b
f
(mn
 m n) VV sin
2
6
if
&lt;
2
],
and analogous expressions for y[ and z[. trary function of v, and ^ be given by
(46)
Hence,
be an arbi
+=(^f^ J
COS
(/&gt;
the functions #
with
# z obtained from (44) by quadratures, together m, n from (42), determine a ruled surface with the linear element (40).
,
, ,
/,
Each choice of gives a different directorcone, which is deter mined by the curve in which the cone cuts the unit sphere, whose center is at the vertex of the cone. Such a curve is defined by a = 0, so that instead of choosing $ arbitrarily we relation
(/&gt;
/(c/&gt;,
&lt;fr)
may
/(&lt;,
take
ifr)
/
0,
by combining equations (43) and and ^r as functions we obtain the expressions for
as arbitrary; for,
&lt;
of
v.
Hence
:
A
ruled surface
may
be
deformed in such a way that
the director
cone takes an arbitrary form.
* Crelle, Vol. XVIII (1838); pp. 297302.
RULED SURFACES
When
the given ruled surface
is
345
nondevelopable, the radicand
in (44) is different from zero, and consequently there are two dif ferent sets of functions XQJ yQ1 Z Q Hence there are two applicable
.
ruled surfaces with the same directorcone.
distribution of these
are
If the
parameters of
found
:
to differ
only in sign.
two surfaces be calculated by (VII, 73), they Hence we have the theorem of
Beltrami *
A
ruled surface admits of an applicable ruled surface such that
corresponding generators are parallel, and the parameters of distri
bution differ only in sign.
144. Particular deformations of ruled surfaces.
By means
of the
preceding results
we prove
the theorem
:
ruled surface may be deformed in an infinity of ways so that a given curve becomes plane.
A
surface.
Let the given curve be taken for the directrix of the original Assuming that a deform of the kind desired exists, we
its
take
plane for the zyplane.
a n cos
2
From
I
(44)
we have
2
+
bn
f
f
(lm
m)
a sin
2
6
2
=
2
0,
which, in consequence of (42) arid
b
cosc/&gt;.(
(43),
2
reduces to
2
&lt;//
+a
2
sine/)
cos#
cosc^Va
Va
2
sin
b2 =
0.
The
integral of this equation involves an arbitrary constant,
is
and
thus the theorem
proved.
to the class of problems
The preceding example belongs
general statement
is
whose
as follows:
into a ruled surface in such a
To deform a ruled surface
the
way
that
deform of a given curve C on the original surface shall possess a certain property on the resulting surface.
We
consider this general problem.
Let the deform of C be the
; , , ; ,
J TW O n X /* , v directrix of the required surface, and let , /3 , 7 denote the directioncosines of its tangent, principal normal, and
If denotes the angle between the osculating plane to the curve and the tangent plane to the surface, we have
binormal.
&lt;r
(46)
I
=#
cos #
*
f
sin
(/
cos
&lt;r
+X
sin
&lt;r),
Annali, Vol. VII (1865), p. 115.
346
DEFORMATION OF SURFACES
and similar expressions for m and n. When these values are sub stituted in the first two of equations (41), the resulting equations are reducible, by means of the Frenet formulas (I, 50), to
coscr
/,
b
,
P ~cos0 n
(47)
(cos
cr
.
sin
n
. ,,
.
sincr sin# sin cr sin
2
~]
)
H
+
T/ (sin
cr
sin
aM
Q)
COS
cr
sin
ft"]
\ab.
I2
;
2
These are two equations of condition on cr, /o, T, as functions of v. Each set of solutions determines a solution of the problem for, the directrix is determined by expressions for p and r, and equa
tions (46) give the directioncosines of the generators. leave it to the reader to prove the above theorem
We
by
this
means, and
we proceed
to the proof of the theorem:
A
curve
ruled surface may be deformed in such a manner that a given C becomes an asymptotic line on the new ruled surface.
On
the deform
we must have a =
or
a
= TT,
so that from (47)
p
the sign being fixed by the fact that p second of (47) reduces to
2
6&gt;
is
necessarily positive.
The
sin
If the
curve with these intrinsic equations be constructed, and in the osculating plane at each point the line be drawn which makes
with the tangent, the locus of these lines is a ruled the angle surface satisfying the given conditions. When the curve C is an orthogonal trajectory of the generators,
the
same
is
true of
surface
its
deform.
be
Hence
:
A ruled
ators become the
deformed in such a way that all the gener one of their principal normals of the deform of any
may
orthogonal trajectories.
in Having thus considered the deformation of ruled surfaces which the generators remain straight, we inquire whether two
RULED SURFACES
347
ruled surfaces are applicable with the generators of each corre sponding to curves on the other. Assume that it is possible, and
let v
= const,
be the generators of
on S corresponding to the
S and u = const, generators of Sr From
the curves
(V, 13)
it
follows that the conditions for this are respectively
where
K=
\/p\
But equations
(48)
are
the
necessary and
applicable to
sufficient conditions that there be a surface
2
S
and Sv upon which the asymptotic lines are parametric (cf. VI, 3). But the curves v = const, and u = const, are geodesies on S and 8^ and consequently on 2. Therefore 2 is doubly ruled. Hence
:
If two ruled surfaces S and Sl are applicable to one another, the generators correspond unless the surfaces are applicable to a quadric with the generators of S and Sl corresponding to the two different
systems of generators of the quadric.
EXAMPLES
1.
A
A
ruled surface can be deformed into another ruled surface in such a
way
that a geodesic becomes a straight line.
2.
a right conoid the latter Prove the converse also.
;
ruled surface formed by the binomials of a curve C can be deformed into is the right helicoid when the torsion of C is constant.
3.
On
the hyperboloid of revolution, defined
xwu.v = +
c
by
v
c
cos c
sin c
,
y c
A
,
=
u
.
sin

A
cos c
v
,
z
= u
d
.
A
the generators
where A 2 = c 2 + d 2 the under the anle cos4.
circle of
gorge
is
a geodesic, which
is
met by
Show
that the ruled surface which results from the deformation of the
3, in
hyperboloid of Ex.
which the
cos u
,
circle of gorge
becomes
z
straight,
v.
is
given by
x
5.
=
ud
Ad
d2
.
ud
y
Ad
.
sin

v
,
= A
uc
\
Show
x c
with parallelism of corresponding generators
v = u cos  H
that the ruled surface to which the hyperboloid of Ex. 3 is the helicoid
is
applicable
A
c
c2 C2
+
d2
v sin i
c
y c
= u sin v
.
A
c
+
c2
d2
cos 
v
,
z 
=
u
I
2 c

c2
d2
c
d
A
A2
v,
and that the
6.
circle of
gorge of the former corresponds to a helix upon the latter.
is
When
the directrix
Bin * .*;
a geodesic, equations
6
(47)
reduce to
+
=
0,
348
7.
DEFORMATION OF SURFACES
When
an hyperboloid
of revolution of one sheet is
deformed into another
ruled surface, the circle of gorge becomes a Bertraml curve and the generators are parallel to the corresponding bmormals of the conjugate Bertrand curve.
8.
A
ruled surface can be deformed in such a
of arbitrary radius.
way
that a given curve
is
made
to lie
upon a sphere
9.
When
a ruled surface admits a continuous deformation into
is
itself
the total
curvature of the surface
constant along the line of striction, the generators meet the latter under constant angle, and the parameter of distribution is constant (cf. 126).
10. Two applicable ruled surfaces whose corresponding generators are parallel cannot be obtained from one another by a continuous deformation.
GENERAL EXAMPLES
1. Determine the systems of coordinate lines in the plane such that the linear element of the plane is ^ U 2 _j_ ^2
=
where
2.
3.
U and V are
functions of u and u respectively.
1.
Solve for the sphere the problem similar to Ex.
Determine the functions 0(w) and ^
x
(u) so that the helicoids, defined
by
= a\/U 2
6
2
cos,
y
=
shall be applicable to the surface
whose equations are
where
4.
U is any
Apply
function of
u.
the
method
of Ex. 3 to find helicoids applicable to the pseudosphere
;
to the catenoid.
5.
The equations
x
=
a V2 u
2 cos 
,
a
define a paraboloid of revolution.
y
=
a
V2
it
2 sin
,
z
=
a
(u 2
it
1)
Show
/*
that surfaces applicable to
are defined by
,
X

id r
/3 02 ~/203 + J
(fzdfz fsdfz)

/*
J
(02^03
~
03 dfa)
?
2
= I
/201
/102
where a
/3
is a real constant, and the respectively such that
/
s
and
s
are functions of a parameter
a and
6. Investigate the special case of Ex. 5 for which functions, and 2+ /2 = l fl = 7=
a and /3 are conjugate imaginary
a2a* 2V2a
.2a2a* 2V2a
,
the/
/ =*
s.
and the
s
are functions conjugate imaginary to
GENERAL EXAMPLES
7.
349
Show
that the surface of translation
x
is
a(cosw
+
cosv),
y
a(sinw
+
sinv),
z
=
c(u
+ v)
applicable to a surface of revolution.
8. Show that the minimal surfaces applicable to a spiral surface (Ex. 22, p. 151) = cio m ~ in , and that the asso are determined by the functions F(u) = cu m + in one. ciate surfaces are similar to the given
,
4&gt;(u)
9.
If the coefficients
E, F,
6?
of the linear element of a surface are
homogeneous
functions of u and v of order
2,
the surface
is
applicable to a surface of revolution.
10. If z, y, z are the coordinates of a surface
S
referred to a conjugate system,
the equations
ctf__ ~
dx aw
if
~ ~ W__pdy_ ^i_p^:. ^LQ^L ^Q^y.
aw au
aw
aw~
aw
au
au
au
aw
cv
~ = Q~
au
are integrable
P and
Q
satisfy the conditions
Show
where the Christoffel symbols are formed with respect to the linear element of S. that on the surface S whose coordinates are x y z the parametric curves form a conjugate system, and that the normals to S and S at corresponding points
, , ,
,
are parallel.
11.
Show
that for the surface
x

f\fi(u)du
is
+
0i(w),
y
= f A/2 (w)dit +
u,
2 (u),
z
=
,
j \f
3
(u)du
+
3 (u),
where \
any function
u respectively, the parametric curves
of Ex. 10 to this
u and /2 /3 0i, 2 form a conjugate system. Apply the results of v. surface, and discuss the case for which X is independent
of
u and
and/!,
,
;
3 *are functions of
12. If S and Si are two applicable surfaces, and S{ denotes the surface corre the same sponding to Si in the same manner as S to S in Ex. 10 and by means of functions P and Q, then S and S{ are applicable surfaces.
13. If x,
?/,
z
and
i, 2/1,
z\ are the coordinates of
a pair of applicable surfaces
S and
Si,
a second pair of applicable surfaces S and S{ is denned by  h(z + zi) + k(y + x = x + h(z + zi)  k(y + T/J), x[ = x l
2/1),
y
z
= y + k(x + xi)g(z + *i), = z + g(y + yi)  h(x + x
^,
yi
zi
x ),
= =
z/i
zt
 k (x + xi) + g(z +  g (y + z/i) + h(x +
;
zi),
KI),
where
and fc are constants. Show that the line segments joining correspond S and S are equal and parallel to those for Si and S{ that the lines for S and S{ and joining corresponding points on S and Si meet the similar lines that the common conjugate system on S and Si corresponds to the common conju gate system on S and Si.
#,
ing points of
;
14.
Apply the
results of Ex. 13 to the surfaces of translation

x
=w 2
v2
2 av,
y
2
=
2
w2
I
v2

2 au
 2* V& 2 + 3 w2 dw,
z
=
2 6u,
2 u2

2 au

Z!
=
2
1,
fa
2
 3u 2 dv.
S
is
Show
that
when g = h =
0,
k
=
the surface
an
elliptic paraboloid.
350
15.
DEFORMATION OF SURFACES
Show
that the equations
"
"
2
"a
y
~
J
where the accent indicates differentiation with respect to the argument, define a family of applicable surfaces of translation. Apply the results of Ex. 12 to this case.
16.
Show
that
when S and
their generating curves correspond, the
Si in Exs. 12 and 13 are surfaces of translation, and same is true of S / and S{.
17. If lines be drawn through points of a Bertrand curve parallel to the binormals of the conjugate curve, their locus is applicable to a surface of revolution.
18. If a real ruled surface
is
to the right helicoid or to a hyperboloid of revolution of
applicable to a surface of revolution, it is applicable one sheet (cf. Ex. 9, 144).
19.
A
ruled surface can be deformed in an infinity of
ways
so that a curve not
orthogonal to the generators shall be a line of curvature on the new ruled surface, unless the given curve is a geodesic in the latter case the deformation is unique
;
and the
line of curvature is plane.
20. Let
P be
any point
of a twisted curve C,
and MI,
/
{
M
2
points on the principal
normal
to
C
such that
=  PM2 =
is
/
/f
a sin (
H
where
a, 6
are constants and p
the radius of curvature of C.
The
loci of the lines
through
21.
M\ and 3f2
parallel to the tangent to
C
at
P are
00
applicable ruled surfaces.
On
the surface whose equations are
x
=
M,
y
=f(u)&lt;f&gt;
(v)
+
i//(v),
z
=
/(u)[0(i&gt;)
(u)]
+ t(o)
fl^ (u),
the parametric curves form a conjugate system, the curves u = const, lie in planes const, in planes parallel to the xaxis parallel to the yzplane, and the curves v
;
hence the tangents to the curves u curve v = const, are parallel.
=
const, at their points of intersection with a
22. Investigate the character of the surfaces of Ex. 21 in the following cases = Vv2 l (b), (u) = const. (c), t(v) = Q; (d), f(u) = au + b. (v) (a),
:
+
;
;
23. If the equations of Ex. 21 be written
the most general applicable surfaces of the same kind with parametric curves cor responding are defined by
where
AC
is
a parameter, and the functions
4&gt;i,
$2, ^i,
^2
satisfy the conditions
$2
+
&lt;I&gt;
= 02 + 0 
K,
4&gt;i*
+
&lt;J&gt;
2
2
=
0{2
+
02,
~
$1(0212
Show
also that the determination of
4&gt;i
and
3&gt;
2
requires only a quadrature.
CHAPTER X
DEFORMATION OF SURFACES. THE METHOD OF WEINGARTEN
145.
Reduced form of the linear element. Weingarten has re
of all surfaces appli
marked that when we reduce the determination
cable to a given one to the solution of the equation (IX, 18),
(1)
namely
J = (\\6)K,
we make no
use of our knowledge of the given surface, and in reality are trying to solve the problem of finding all the surfaces with an assigned linear element. In his celebrated memoir, Sur la
deformation des surfaces,* which was awarded the grand prize of the French Academy in 1894, Weingarten showed that by taking
by another which can be solved
chapter
is
account of the given surface the above equation can be replaced in several important cases. This
begin by determining a particular moving trihedral for the given surface. It follows from (VII, 64) that the necessary and sufficient con dition that the directrix of a ruled surface be the line of striction is
(2)
devoted to the exposition of this method.
We
6
= a#
+#X+*X=0.
We
(2)
The functions // m/ n are proportional to the directioncosines of the curve in which the directorcone of the surface meets the unit
sphere with center at the vertex of the cone. the spherical indicatrix of the surface. From
ll
call this
curve
and the identity
+mm + nn =
seen that the tangent to the spherical indicatrix is perpen dicular to the tangent plane to the surface at the corresponding point of the line of striction. This fact is going to enable us
it is
under what conditions a ruled surface 2, tangent to a curved surface S along a curve C, admits the latter for
to determine
its line
of striction.
*Acta Mathematica, Vol.
351
XX
(1896), pp. 159200.
352
DEFORMATION OF SURFACES
suppose that the parameters
is
We
w, v are
any whatever, and that
the surface
referred to a
moving
trihedral.
We
consider the
ruled surface formed by the zaxis of the trihedral as the origin of the latter describes the curve C. The point (1, 0, 0) of a second
trihedral parallel to this one, but with origin fixed, describes the spherical indicatrix of 2. From equations (V, 51) we find that the
components of a displacement of
0,
this point are
r du 4 r^v,
(qdu
+ q^dv).
In order that the displacement be perpendicular to the tangent plane to 2 at the corresponding point of (7, that is, perpendicular to the zyplane of the moving trihedral, we must have
(3)
rdu
if
+
r l dv
= Q.
Hence
the
a trihedral
ner, as the vertex of
2&gt;axis
T be associated with a surface S in any man T describes an integral curve of equation (3),
a ruled surface
of
T generates
whose
line of striction is
this curve.
When
the parametric lines on
a&gt;axis
S
which the
of
T makes with the
are
are given, and also the angle tangent to the curve v const.,
U
the functions r and r l are completely determined, as follows from
(V, 52, 55).
They
Hll
Hence
if
cU
U be
given the value
C121
///
&lt;f&gt;
It")
*+*&lt;&gt;
where (u) denotes an arbitrary function of M, the function r is zero, and as the vertex of the trihedral describes a curve u = const., the zaxis describes a ruled surface whose line of striction is this curve.
t
= Suppose now that the trihedral is such that r x 0. From (V, 48, 64)
it
follows that
(6)
consequently
(1)
r= C
ty is
where
an arbitrary function of
u.
PARTICULAR TRIHEDRAL
353
Let the righthand member of (7) be denoted by f(u, v), and change the parameters of the surface in accordance with the equations
u =u,
l
v^
= f(u,
v).
From
32 and equation
(7) it
follows that
idv
unaltered by the transformation, in terms of the new coordinates is equal to V unity, and hence from (6) we have
Since
is
K
HK
Therefore the coordinate curves and the moving trihedral of a surface can be chosen in such a way tnat
r
(8)
= vr
^=0,
we
It should be
r
= v,
HK=l.
In this case
reduced form.
say that the linear element of the surface is in its remarked that for surfaces of negative
curvature the parameters are imaginary.
146. General formulas. If X^ Y^ Z^ X, r, Z denote A;, F2 the directioncosines of the axes of the moving trihedral with
; ,
Zj&gt;
respect to fixed axes,
we
have, from (V, 47),
du
(9)
il
dv
"
_ Xa qi
q&lt;&gt;

dv
$i satisfy
dv
The
rotations p,p^
equations (V, 48) in the reduced form
dv
du
x, y, z
dv of
du
reference to these fixed axes are
The
coordinates
c
S with
given by
/
(11)
y
2
= = f(^i + ^2) ^ + (f 1^1 +
where
and
(13)
^
2)
^N
dv
du
"
354
DEFORMATION OF SURFACES
s
Weingarten
f*
77,
,
method
consists in replacing the coefficients of
of
u
f t rj l in the last of equations (13) by differential parameters formed with respect to the linear element of the spherical
representation of the zaxis of the
moving
is
2
)
trihedral.
By means
(14)
of (9) this linear element
reducible to
da 2 =
dX + dY + dZ = (v + q
2
2
2
2
du2 + 2 qq^dudv
+q
2
dv
2
.
The
differential
parameters of u, formed with respect to this
* form, have the values I
^
(15)
_4(y2 +
v *ti
2
2
)
A
(u*
A
u}
=
Aq
v^q l
Aoit
=
q
v ql
p
vq l
Because of the identity (V, 38)
we have
(16)
also
A^=
2
^by q lt and the values of and (16) be substituted, we have
If the last of equations (13) be divided
i
Pi/2i obtained from (15)
22
3
2
4
t
i
2v
In consequence of the
(18)
first
of equations (15), written
v
= L=,
VAjt*
in (17) are expressible in terms of to (14), as was
the coefficients of
f,
TJ,
fv
,
rj l
differential parameters of
u formed with respect
to be proved.
0. Under this condition An exceptional case is that in which the spherical representation of the zaxis reduces to a curve, as is seen from (14).
q^
*
Previously
we have
indicated
to the linear element of the spherical representation. regard this practice in this chapter.
by a prime differential parameters formed with respect For the sake of simplicity we dis
THEOREM OF WE1NGAKTEN
By means
(19)
355
of (9)
we
find that
A, (A,, tO
=
^.
M^,
(11)
)
=
^f.
\(Z
)
=
f
and consequently equations
(20)
may
du
be written
x
=[^ + W\ (X
*&gt;)]
+ [f
^ + v,v\ (X
u)] dv,
and similarly for y and 2. 147. The theorem of Weingarten. Equation (17) is the equation which Weingarten has suggested as a substitute for equation (1).
We
notice that
f,
?;,
fx
,
T/ I
are
known
functions of u and v
when
of (18) equation (17) can be given. a form which involves only u and differential parameters given of u formed with respect to (14). On account of the invariant
is
the surface
S
By means
of these differential parameters this linear element be expressed in terms of any parameters, say u and v may We shall show that each solution of equation (17) determines a surface applicable to S. We formulate the theorem of Wein
.
character
garten as follows
:
Let S
(21 )
be a surface
whose linear element in the reduced form
)
is
ds*
%,
= (? +
2
?; 1
2
T?
du* + 2
(^ + wj dudv + (tf +
u and
v such that
then
rj,
fj,
are functions of
Z^ le the coordinates of a point on the unit in terms of any two parameters u and v , the linear sphere, expressed
Furthermore,
let
Xv Yv
2
1
element of the sphere being
(23)
da
integral
1
=
& du
*
+2&
du dv
+
&gt;dv
\
Any
(24)
u of
l
the equation
Ju,
L
A M u  Ju, l=

u,
A,w
t
)=
0,
356
the differential
DEFORMATION OF SURFACES
parameters being formed with respect
to (23),
renders
the following expression
and similar ones in y and
z total differentials:
(25)
where
f Ae surface whose coordinates are the functions has the linear element (21).
x, y, z
thus defined
and
Before proving this theorem we remark that the parameters u v may be chosen either as known functions of u and v, or in
that the linear element (14) shall have a particular In the former case X^ v Z^ are known as functions of u
such a
form.
way
Y
f
and
v
,
and
in the second their determination requires the solution
of a Riccati equation.
However,
r
f
in
what follows we assume that
are known. l Suppose now that u and v are any parameters whatever, and that we have a solution u^ of equation (24), where the differential
Xv
Yv Z
parameters are formed with respect to (23).
quantity
(A^)"*.
Let
v^
denote the
v
,
Both
u^
and v l
are functions of
u and
and
consequently the latter are expressible as functions of the former. We express X^ Y^ Z{ as functions of u and v l and determine the
l
corresponding linear element of the unit sphere, which
(26)
l
we
write
dffl
= (; dul +
l
2
^ dujvt + ^
c(v*.
In terms of u and v we have
,
i
From
these expressions
it
follows that
if
we put
we have
(
27 )
METHOD OF WEINGARTEN
Hence
if
357
we put
x = Y&  Z,Y
the functions A^,
tions (V, 47).
Y = z&  x&,
.
z =x Y  r^,
1
a
Yv
,
Z satisfy a set of equations similar to
equa
In consequence of (27) the corresponding rotations have the values
dX
.dx
r1=
It is readily
0.
dX
shown
that these functions satisfy equations similar
to (10).
Since the functions
f,
in (21), equations similar to the first
sarily satisfied.
Hence
same form in (25) as of equations (13) are neces the only other equation to be satisfied, in
77,
,
^
are of the
two
order that the expressions (25) be exact differentials,
is
But
it
can be shown that the coefficients of (26) are expressible in
the form
so that
g _ v *\ ^
by means
&lt;%
__
^_
2
of differential parameters of u^
formed with
respect to (26) the equation (28) can be given the form (17). Hence all the conditions are satisfied, and the theorem of Wein
garten has been established.
148. Other forms of the theorem of Weingarten. It is readily found that equations (22) are satisfied by the expressions
dv
(29)
du dv
dv
v.
where
(/&gt;
is
any function of u and
,7
Since
now
(30)
+ ^=0,
equation (17) reduces to
358
DEFOKMATION OF SURFACES
of
This equation will be simplified still more by the introduction two new parameters which are suggested by the following
considerations.
As previously defined, the functions X^ v Z^ are the directioncosines of lines tangent to the given surface S in such a way that the ruled surface formed by these tangents at points of a curve u = const, has this curve for its line of striction. Moreover, from
the theorem of Weingarten it follows that the functions X^ Yv Zl have the same significance for the surface applicable to S which
Y
corresponds to a particular solution of equation (17). But v Yv Zl may be taken also as the directioncosines of the
X
normals to a large group of surfaces, as shown in 67. In partic ular, we consider the surface S which is the envelope of the plane
$+
Each solution
Zj,
= u.
of equation (17) determines such a surface. If x, y, z denote the coordinates of the point of contact of this plane with S,
we have from
(32)
(V, 32)
x
= uX +\(u,X
l
1 ),
which, in consequence of (19),
(32
may
v
be written
)
i = wXfX.
of contact of S lies in the plane through the origin to the tangent plane to S at the corresponding point. parallel If the square of the distance of the point of contact from the
Hence the point
origin be denoted
by 2
a
^,
and the distance from the origin
a a
to the
tangent plane by p,*
(33)
we have
2g = s
(V, 35, 37)
+ya +i =w +^.
p=u
.
From
it
follows that the principal radii of
2
are
given by
(34)
*
The reader
will observe that the functions
the rotations designated by the same letters. the treatment of the theorem of Weingarten, risk of a confusion of notation.
p and q thus defirfed As this notation is
are different from
it
in generally employed has seemed best to retain it, even at the
METHOD OF WEINGARTEN
where the
differential parameters are
359
(14).
formed with respect to
From
(35)
these equations
we have
We
shall
now
effect a
change of parameters, using
ones.
^*r o_
defined by (33) as the
y^r
new
*^r
^
I
By
^
direct calculation
^*r ^
v *r
.;?
p and q we obtain
.
du
o

_
i
dp
dq
dv
.

o
v*
dq
dif
+P
dp*
2
a/?a^
z. /^
^
T 2
+
~
a^
(36)
_i
cudv
vdpcq
Q^,2r
^1^ ~
dv
2
1
tf
v
6
2
dq
By means
of the equations (33)
and
(36) the
fundamental equa
tion (31) can be reduced to
(37)
This
is
the form in whicli the fundamental equation was
first
con
sidered by Weingarten.*
The method
of
146, 147 was a subse
quent development. In terms of the parameters
p and
q the formulas (29)
become
dpdq
(38)
dq
If these values
and the expression
it is
for
\(u, X^) given by
(32) be
substituted in (20),
reducible to
"
dp*
*
dpdq/
\
"tip
tiq
cq
Comptes Rendus, Vol. CXII
(1891), p. 607.
360
DEFORMATION OF SURFACES
for
Hence the equations
S may
be written
(39)
and consequently the linear element of S
is
of the
form
from those which figure
Since these various expressions and equations differ only in form in the theorem of Weingarten, the latter is
remark also that the rightjust as true for these new equations. hand member of (40) depends only upon the form of c. Hence we
have the theorem of Weingarten in the form
:
We
a definite function of p and q, this with the same spherical equation defines a large group of surfaces the functions p l and p 2 denoting the principal radii, representation,
When
(j)
in equation (37)
is
and p and 2q
the distance
from
the origin to the tangent
plane and
surface
the square of the distance to the point of contact.
Each
2
a surface with the satisfying this condition gives by quadratures (39) each surface with this linear element linear element Conversely,
(40).
stands in such relation
equation (37).
to
some surface satisfying the corresponding
As
a corollary to the preceding results,
we have
the theorem
:
The linear element of any surface S
(41)
di~
is reducible to the form
=
du*
+
2
^ dudv + ^ dv\ du
2
dv
v.
where
^r is
a function of u and
that the linear element of any surface is reducible to the form (40). If, then, we change the parameters by
For,
we have seen
means
of the equations
we have
(43)
ds*
= du*+2p dudv + 2q dv\
METHOD OF WEINGARTEN
From
(42) it follows that
2 2 2 ,
4&gt;
361
&lt;fu
=a
^ 1
dp +
,
3
i&lt;t&gt;
,
dt&gt;
=
tf$ i
,
&lt;fy,
dp
+
,
d
dp
ejpdg
dp 3
a/
^
rfg,
and consequently
A
(44)
A
where
2 2
=
dp
dq
\dp dq
the inverse
From
~(44) it is seen that dv
=^j and consequently cu
of equations (42) are of the
fAfii
form
(45)
= X,
du
(43) is of the
W
q *
= ^L.
dv
(41), as
W
Hence equation
form
was to be proved.
Moreover, equations (44) reduce to
(46)
2
=
A
=
(#?/
 A
dpdq
w, v
In terms of these parameters
(47)
equations (39) reduce to dz
dx
= X^du + ^c?v,
of
= I^c^tt + ^^v,
are given
= Z^du f ^c?v.
Hence the coordinates
AQ
(48)
2
by
J
_
X
^C = T^
dv
y = dv
_
3v
z
=
^2
f
^" dv
and the directioncosines
(49)
of the
normal to 2 are
X.A
aw
is,
r = ^,
l
aw
Z1==^,
^
that
the normals to
2
are parallel to the corresponding tangents
to the curves v
= const, on S. Hence we have the
the
following theorem
:
When the linear element of a surface is in face 2 whose coordinates are given by (48)
form
(41), the sur
has the same spherical
362
DEFORMATION OF SURFACES
its
representation of
normals as the tangents
to the
curves v
= const.
on
S. If p and 2q denote the distance from the origin to the tangent plane to S and the square of the distance to the point of contact, they
have the values (45).
Moreover, if the change of parameters defined
by these equations be expressed in the inverse form
/cn (50)
,
M
=
d&lt;f&gt;
dp
the principal radii of
s
v=
d(f&gt;
*
dq
2
satisfy the condition
and
(52)
the coordinates of
S
are given by quadratures of the form
dx
with the same representation as 2, and whose determines by equa functions p v /? 2 p, q satisfy (51) for the same tions of the form (52) a surface applicable to S.*
Moreover,
ever// surface
,
(f&gt;,
149. Surfaces applicable to a surface of revolution.
When
the
linear element of a surface applicable to a surface of revolution
is
written
(53)
d?=du* + p*(u
zaxis of the
l
)dv*,
is
and the
v
moving
is
trihedral
tangent to the curve
const., the function r
equal to zero, as follows
(8),
from
(4).
In order to obtain the conditions
of variables
we
effect the transformation
u
= v^
ds*
2
v
=
u^
so that the linear element becomes
(54)
= p du*+dv
2
.
Now
(55)
r
=p
f
,
element in
7^=0, and consequently in order the reduced form we must take u
to
have the linear
= u,
v=p (v).
* For a direct proof of this theorem the reader is referred to a memoir by Goursat, Sur un theoreme de M. Weingarten, et sur la the orie des surfaces applicables, Toulouse Annales, Vol. V (1891) also Darboux, Vol. IV, p. 316, and Bianchi, Vol. II, p. 198.
;
SURFACES OF REVOLUTION
From
surface
these results and (32
)
363
we
find that the coordinates of the
2
are given
by
p cu v
f
p du^
p dv l
p dv l
i==
_lJl + !!l.^,
p cu^
p d Vl
to
and the directioncosines of the normals
.A.,
2
~
Zs*
are
1 fo Y = 
&gt;
JL
v
1
1 fy 
5
1 8* = 
p dv l
p cv l
2
p dv 1
Also,
(56)
we have
P
=^xX^v v
:
2?
=^ =^ + ~
Hence we have the theorem
To a curve which
i*
the
deform of a meridian of a surface of revo
planes
origin,
lution there corresponds on the surface 2 a curve such that the tangent to 2 at points of the curve are at a constant distance from the
a deform of a parallel there corresponds a curve such that the projection of the radius vector upon the tangent plane at a
to
and
point
is
constant.
For the present case
77
=f =
1
;
consequently
we
have, from (38),
Sf
This equation
(57)
is satisfied
by
2
&lt;/&gt;(&gt;,
#)=/(2
q
jt?
),
where
/ is
any function whatever.
In terms of this function
we
have, from (38),
where the accents indicate differentiation with respect to the
argument,
2qp
2
.
By means
of (55) the linear element (54) can be transformed into
the function
a)(v)
being defined by
364
Since
77
DEFORMATION OF SURFACES
=
fl
=
0,
we have
and we know that
r
= v. Now
equations (58) become
and these are consistent because of the relation 2^ p~ = \/v 2 which results from (56). Hence we have the theorem
:
,
When
is
&lt;
(p, q) is a function of 2 q
to
applicable
p\ the corresponding surface S a surface of revolution, the tangents to the deforms of
the form (57) and put
2
the parallels being parallel to the corresponding normals to 2.
If
we
give
&lt;j&gt;
^ = 2f,
2 2
the linear element
of
Sis
ds
(59)
= (^q
2 2 p ) d^ +
^fr
dp
,
as follows
from (40) or (58). 150. Minimal lines on the sphere parametric. In 147 we re marked that the parametric curves on the sphere may be any what ever. An interesting case is that in which they are the imaginary
In
generatrices.
35 we saw that the parameters of these
lines,
say a and
(60)
/3,
can be so chosen that
rp
X,= + a/3

a/31
a/3
Consequently
(61)
da 2 =dX? +
(32)
4
dad /3
From
are
we
find that the coordinates of 2, the envelope of
the plane
Xx + Yy+Zz
p
=
(62)
z
=
From
(63)
these
we
obtain
2
MINIMAL LINES ON THE SPHEKE
By means of p and its
found, and
365
of (34) the expressions for p l in terms p 2 and derivatives with respect to a and yS can be readily thus the fundamental equation (37) put in a new form.
+
p^
not with the general case that we shall now concern but with a particular form of the function ourselves, q). This function has been considered by Weingarten * it is
However,
it is
&lt;j&gt;(p,
;
(64)
In this case
so that equation (37) reduces to
(65)
/&gt;i
+ ft =
(2;&gt;
+
be written
(P)
which, in consequence of (34),
(66)
may
dad j3
(l
+ a/3)
2
When
the values from (62) are substituted in (52),
we
obtain
(67)
z
= *&  Cu dZ +
J
1 l
\
occ
where
(68)
From
(42)
and
(64)
we have
u
qp
2
2
a&gt;
f
(
p),
v=p.
in this case,
Hence
(69)
the linear element (43) of
ds
2
*S is,
=du +2v dudv +
2 [u
+ v + w (v)] dv
2
2
.
*Acta Mathematica, Vol.
XX
(1896), p. 195.
366
DEFORMATION OF SURFACES
(68)
it is
However, from
(70)
seen that
Ul
, =u + v
2
so that (69)
(71)
may
be written
ds
2
=
dul + 2 [ MI +
a
*&gt;
.
(t&gt;)]
theorem Gathering together these results, we have the
2%e determination of
reduces
to the
:
all the surfaces ivith the linear
element (71)
integration of the equation
for o(_p) arbitrary integral of this equation However, the integral is known in certain cases.
The
is
not known.
consider
We
several of these.
151. Surfaces of Goursat.
oloids.
Surfaces applicable to certain parab
When we
take
(73)
v (p)=im(lm)p\
m
being any constant, equation (72) becomes
m(lm)p
dad ft
(l
+ aj3f
can be found by the method general integral of this equation of Laplace,* in finite form or in terms of definite integrals, accord
The
ing as
m
The
(75)
integral or not. linear element of the surface
is
S
is
ds
1
= du? + [2 u^ H m (1  m) ^] dv\
are such that
And
(76)
the surfaces
2
p 1 + pt
is,
+2p = m(mI)p,
of the principal radii is proportional to the dis tance of the tangent plane from a fixed point. These surfaces were first studied by Goursat, f and are called, consequently, the
that
the
sum
surfaces of G our sat.
*Darboux, Vol.
II, p. 66.
t
American Journal, Vol.
X
(1888), p. 187.
SURFACES OF GOURSAT
367
Darboux has remarked* that equation (71) is similar to the linear element of ruled surfaces (VII, 53). In fact, if the equations of a ruled surface are written in the form
(77)
x
,
= x +lu
;
l,
y
w, n are functions of v alone, which now is not necessarily the arc of the directrix, the linear element of the surface will have the form (71), provided that
where #
/,
(78)
2J 2 = 1,
2a^ =
0,
2X = 2 w (^), 2a# = l,
2
2/
2
=0.
In consequence of the equations
2ft
it
=0,
follows that a ruled surface of this kind admits an isotropic = 0, that is, if plane director. If this plane be x + iy
we have
where
V is
a function of
v.
By means
form
*dv
of these values
and equa
tions (78),
we can put
(77) in the
(79)
= %Vu^+% Cv v dv 
C~ dv,
/y
We
shall find that
dv

yt
among these surfaces there is an imaginary which are applicable certain surfaces to which Weingarten called attention. To this end we consider the function
paraboloid to
_2p^
(80)
(_p)=:
^ficp
2
tee
v
"
where K denotes a constant.
Now
A
equation (66) becomes
_2
Vic
_
1
* Vol. IV,
p. 333.
368
DEFOKMATION OF SURFACES
In consequence of the identity
the preceding equation
is
equivalent to
dad/3
If
log(l
+ a/3)V^ = JL
we put
this equation takes the Liouville
form
20
&gt;
dad IB
of
which the general integral
is
1
AS
and
/3
where
A
and
B
are functions of a
respectively,
accents indicate differentiation with respect to these.
and the Hence the
general integral of (81)
c
is
%=
VAB (l + a/3)
is
l
2^ + AS)
and the linear element of S
(82)
If
ds
2
= du^2\u  V^K  2
to (80),
now, in addition
we take
V
the equations (79) take such a form that
(83)
(x+iy)x
=
icz.
Hence the surfaces with the linear element (82) are applicable to the imaginary paraboloid (83). The generator x + iy = Q of this paraboloid in the plane at infinity is tangent to the imaginary
circle at the point (x:y:z = l:i:Q), which is a different point from that in which the plane at infinity touches the surface,
that
is,
the point of intersection of the two generators.
DEFOBMATION OF PARABOLOIDS
Another interesting case
value
(84)
If
2.
is
369
afforded
when
(71)
m
in (73) has the
Then
u&gt;
= (v)
ds
2
v
2
,
and equation
(
becomes
=
duf + 2
Ml
we
take V=v/^/2^c,
we
obtain from equations (79)
from which we
(85)
find,
by the elimination
(z
of
it/)
u l and
.
v,
+
i
2
= K (x
The generator x +
iy
=
in the plane at infinity
on the paraboloid
:
1 i: 0), circle at the point (x: y: z (85) is tangent to the imaginary but the paraboloid (85) is just as in the case of the paraboloid (83),
same point. tangent to the plane at infinity at the
GENERAL EXAMPLES
1 moving trihedral can be associated with a surface in an infinity of ways so that as the vertex of the trihedral describes a curve u = const, the zaxis generates a ruled surface whose line of striction is this curve.
.
A
2. The tangents to the curves v = const, on a surface at the points where these curves are met by an integral curve of the equation
form a ruled surface for which the
3.
latter curve
is
the line of striction.
If the ruled surface
formed by an
its line
locus of the points of contact for deformations of S.
infinity of tangents to a surface S has the of striction, this relation is unaltered by
D, D face with the linear element
4
.
Show
that
if
,
D"
(53),
are the second fundamental coefficients of a sur the equation of the lines of curvature of the
associated surface
S
is
reducible to
Ddii!
+D
;
dvi
V
dui
+
D"d
dui
pp dv\
p
5
.
Show
that the surface
S
associated
by the method
of
Weingarten with a sur
with parallelism of tangent to the surface S complementary to S with respect to the deforms of the planes meridians and that the lines of curvature on S and S correspond.
face
S applicable
;
to a surface of revolution corresponds
370
6.
DEFORMATION OF SURFACES
Show
that
when
has the form
(57), the
equation (51)
is
reducible to
hence the determination of all the surfaces applicable to surfaces of revolution is equivalent to the determination of those surfaces S which are such that if MI and is the projection 2 are the centers of principal curvature of 2 at a point I/, and of the origin on the normal at M, the product NMi 2 is a function of ON.
M
N
NM
7.
Given any surface S applicable to a surface of revolution.
Draw through
a
fixed point O segments parallel to the tangents to the deforms of the meridians and of lengths proportional to the radii of the corresponding parallels, and through
the extremities of these segments draw lines parallel to the normals to S. Show that these lines form a normal congruence whose orthogonal surfaces 2 have the
same spherical representation
faces of the equation of Ex.
8.
6.
of their lines of curvature as
S and
are integral sur
Let
fi
complementary
be a surface applicable to a surface of revolution and S the surface to S with respect to the deforms of the meridians let also S and
;
S be Show
surfaces associated with
that
S and S respectively after the manner of Ex. 7. corresponding normals to S and S are perpendicular to one another,
and that the common perpendicular to these normals passes through the origin and is divided by it into two segments which are functions of one another.
9.
Show
that a surface determined
2q
by the equation
PIPZ
+
K
+
(PI
+ pz)p +
=
0,
where
a constant, possesses the property that the sphere described on the seg ment of each normal between the centers of principal curvature with this segment
K is K in great for diameter cuts the sphere with center at the origin and of radius circles, orthogonally, or passes through the origin, according as K is positive, nega tive, or zero. These surfaces are called the surfaces of Bianchi.
V
10.
Show
that for the surfaces of Bianchi the function 0(p, q)
.
is
of the
form
= V2 q  p 2 +
1
/c,
and that the linear element
of revolution
is
of the associated surface
S
applicable to a surface
Show
ds2
also that according as
/c
=
0,
&gt;
0,
or
&lt;
the linear element of
S
is
reducible
to the respective
forms
,
=
dw 2
+ e2M dv 2
ds 2
= tanh 4 u du 2 + sech 2 w du 2
,
ds 2
=
coth4 u du 2
f
csch 2 u dv 2
.
On account of this result and Ex. 10, p. 318, the surfaces of Bianchi are said to be 0. of the parabolic, elliptic, or hyperbolic type, according as K 0, or 0,
=
&gt;
&lt;
11.
Let S be a pseudospherical surface with
its
linear element in the
form
Find (VIII, 32), and Si the Bianchi transform whose linear element is (VIII, 33). the coordinates x, y, z of the surface S associated with Si by the method of Weingarten, and show that by means of Ex. 8, p. 291, the expression for x is reducible to
x
=

aea (cos 6X1
+
sin 0JT2 ) 4
fX,
scaxis of the
where X\, JT2
,
X are the directioncosines with respect to the
S and
of the
tangents
to the lines of curvature of
normal
to the latter.
GENERAL EXAMPLES
371
12. Show that the surfaces S and S of Ex. 11 have the same spherical represen tation of their lines of curvature, that S is a surface of Bianchi of the parabolic type, and that consequently there is an infinity of these surfaces of the parabolic
type which have the same spherical representation of their lines of curvature as a given pseudospherical surface S.
13. Show that if Si and S 2 are two surfaces of Bianchi of the parabolic type which have the same spherical representation of their lines of curvature, the locus of a point which divides in constant ratio the line joining corresponding points of Si and S 2 is a surface of Bianchi with the same representation of its lines of cur vature, and that it is of the elliptic or hyperbolic type according as the point divides
the segment internally or externally.
14.
When S
is
a pseudospherical surface with
its
linear element in the
(VIII, 32), the coordinates x~i, yi, z\ of the surface of Weingarten are reducible to
Xi
S determined by
the
form method
=
A
(ae
J
a cos 6
+
y sin
6)
X\
z 1?
f
(ae
a sin
77
cos 0} JF2 ,
;
JT2 , 2 , Z 2 are the JTi, FI, directioncosines of the tangents to the lines of curvature of S. Show also that S has the same spherical representation of its lines of curvature as the surface Si with
and analogous expressions for yi and
where
Zx
F
the linear element (VIII, 33).
15. Derive
by means
of
from the equations X2 xXi + yYi + zZ t =p, and (49), the equations (44), (48),
+
2
+
2
=
2 g,
where
16.
x, y, 2
are the coordinates of S.
Show
that the equations for
S
similar to (IV, 27) are reducible to
dudv
cv*
\cu*
z.
cucv
(cf.
and similar expressions
in
y and
Derive therefrom
D"dv
Ex. 15) the equations
D
du
dpdq
du
+
+ r(Ddu +
\dq
2
lYdv)

0,
dp*
cpdq
where D,
17.
D
,
D"
are the second fundamental coefficients of 8.
Show
Ex.
that the lines of curvature on
S correspond
to a conjugate
system on
S
(cf.
16).
1 8.
Show
that for the surface
S we have
dx
dx
~
dX\
plp&lt;
dXi
dp
.
dXi
dq
*
2p
19. Let
eq
dq~
yi
S be
the surface defined by (67) and Si the surface whose coordinates are
Xi
=
x
u\X\,
=
y
WiFi,
z\
z
u\L\.
Show
that Si is an involute of /S, that the curves p = const, are geodesies on lines of curvature on Si, and that the radii of principal curvature of Si are
S and
372
20.
DEFORMATION OF SURFACES
Show
that
when
m
/3
in (73)
is
or
1,
the function
p
is
the
trary functions of
a and
respectively, that the linear element of
sum of two S is
arbi
ds*= dw 12
that
+
2i*idw
2
,
S
is
an evolute of a minimal surface
(cf.
Ex.
19),
and that the mean evolute of
S
is
a point.
21.
Show
that
when
m
in (73)
is 2,
the general integral of equation (74)
is
where /i and /2 are arbitrary functions surface 2 is minimal (cf. 151).
22.
of
a and
respectively.
Show
also that the
Show
that the
mean
evolute of a surface of Goursat
is
a surface of Goursat
homothetic to the given one.
23.
Show
that
when
u&gt;(p)
p=
= ^op 2 then a log(l + a/3) +/i(nr) +/2 (/3),
,
where /i and /2 are arbitrary functions, that the linear element of S
ds 2
is
=
is
du?
+
2 (ui
+
2 aw) du ,
and that the mean evolute of 2
24.
tion
a sphere.
Show
that the surfaces
S
of Ex. 23 are applicable to the surfaces of revolu
S whose equations are
v
.
v
a
where a
25.
is
a
=
J
I
c vu* /~17
a2
ia,
is
an arbitrary constant. Show also that when a
that
=
a paraboloid.
Show
when
the surfaces
tive
;
S
also that the surfaces
are spherical or pseudospherical according as are applicable to the surface
&lt;S
m
z
is
positive or nega
x
+
iy
=
v,
xiy =
1)2
M2
2
TT~~ 2m
mu
=
w
which
is
a paraboloid tangent to the plane at infinity at a point of the circle at
infinity.
CHAPTER XI
INFINITESIMAL DEFORMATION OF SURFACES
152. General problem.
of isometric surfaces
The preceding chapters
deal with pairs
which are such that
in order that
one
may
be applied to the other a finite deformation is necessary. In the present chapter we shall be concerned with the infinitesimal deformations which constitute the intermediate steps in such a
finite
deformation.
,
S and
Let x^y,z\ x y\ z respectively be the coordinates of a surface a surface S\ the latter being obtained from the former by a
If
very small deformation.
(1)
we put
f
x
e
^x + ex^,
y
=y + cyv
z
^z + ez^
where
tions of
denotes a small constant and x v y^ z l are determined func u and v, these functions are proportional to the direction
cosines of the line through corresponding points of these equations we have
S and S
f .
From
dx
*
+
*
dy
+
dz n
= da? + dy + dz +
2
2
2e(dx dx l + dy dy
l
+ dz dzj
If the functions satisfy the condition
(2)
dx dx
l
f
dy dy^ + dz dz l =
r
0,
corresponding small lengths on S and S are equal to within terms 2 of the second order in e. When e is taken so small that e may be
neglected, the surface
S
defined by (1)
is
said to arise from
S by
an infinitesimal deformation of the latter. In such a deformation each point of S undergoes a displacement along the line through it whose directioncosines are proportional to x v y v z r These lines
are called the generatrices of the deformation. It is evident that the problem of infinitesimal deformation
is
equivalent to the solution of equation
373
(2).
Since x v y^ z l are
374
INFINITESIMAL DEFORMATION
v,
functions of u and
surface
they
may be taken
for the coordinates of a
the fact that the tangent to r Equation (2) expresses curve on S is perpendicular to the tangent to the correspond any We say that in this case ing curve on Sl at the homologous point.
S and S
ments.
l
linear correspond with orthogonality of corresponding And so we have:
ele
The problem of
equivalent
the infinitesimal deformation of a surface S is to the determination of the surfaces corresponding to it
with orthogonality of linear elements.
153. Characteristic function.
of these surfaces
We
proceed to the determination
(2)
Sv and
to this
end replace equation
by the
equivalent system
Weingarten
(4)
*
replaced the last of these equations by the two
^ ex Sssw**
fix.
^v
X^w=~* 7/
dx dx l
thus denning a function $, which Bianchi has called the character = V EG F*. istic function ; as usual If the first of equations (3) be differentiated with respect to v,
H
and the second with respect to
M,
we have
* dv du dv
dudv
** fa
uv
fr
$
fa du dv
With
the aid of these identities, of the formulas (V, 3), and of the Gauss equations (V, 7), the equations obtained by the differentia
tion of equations (4) with respect to . reducible to
u and
v respectively are
H
v
*
Crelle, Vol.
H
C
(1887), pp. 296310.
CHARACTERISTIC FUNCTION
Excluding the case where S
&lt;r\
375
is
a developable surface,
*
we
solve these
dx.
equations for
&gt;,^ *l du
2*Xzr anc ^ dv
TT\
dx.
obtain
cu
(5)
u
dv
S.
dv
KH
where
and
K denotes
and
the total curvature of
If
we
solve equations
(3), (4),
v,
(5) for the derivatives of x^
y^
zl
with respect to u
we
obtain
v
cu
CU
(6)
KH
dv dv
dv
KH
function
and similar expressions in y^ and z r Hence, when the characteristic is known, the surface S can be obtained by quadratures. l
Our problem reduces
If equations (5)
therefore to the determination of
&lt;.
be differentiated with respect to v and u respec and the resulting equations be subtracted from one another, tively,
we have
+4
V
1
to du
V 2j
i
fa.
fa.
When
the derivatives of
Jf,
F,
Z
8),
in the righthand
member
are
replaced by the expressions (V,
the above equation reduces to
dv
u
d
du
dv
KH
Bianchi
du
KH
(7) is
H
reducible to
calls this the characteristic equation.
In consequence of (IV, 73, 74) equation
ft
376
INFINITESIMAL DEFORMATION
&lt;",
where
c^,
$
are the coefficients of the linear
/S,
element of the
spherical representation of
(9)
namely
2
da* =
&du* +
/K
^ dudv
reducible to
and
By means
of (V, 27) equation (8)
is
where the Christoffel symbols are formed with respect
Since X, Y,
to (9).
Z
tions of (10), and latter equation may be written
are solutions of equations (V, 22), they are solu consequently also of equation (7). Therefore the
dX
dv
du
KH
du
v
_
But this we have
is
the condition of integrability of equations
:
(6).
Hence
the theorem
Each solution of the S^ and consequently an
characteristic equation determines a surface
infinitesimal deformation of S.
154. Asymptotic lines parametric.
When
the asymptotic lines
on 8 are parametric, equation (10)
(VI, 15), to
is
reducible, in consequence of
dudv
2
dv
du
2
du
dv
where
If
we put
&lt;f,
V ep = 6,
ASYMPTOTIC LINES PARAMETIC
e
377
is
being
+1
or
1 according as the curvature of
S
positive or
negative, equation (11)
(12)
becomes
**
Since X, Y,
vl
Z are = XV
solutions of (11), the functions
e/j,
v2
= FV
ep,
are solutions of (12).
Now
(13)
equations
(6)
may
be put in the form
e
dx,
e
dv
cB
cv
du
du
The reader should compare
formulas
(
79),
which give the expressions
these equations with the Lelieuvre for the derivatives of
i/ i/
the coordinates of
S
in terms of
it
1?
2
,
i&gt;
3
.
From
these results
follows that any three solutions of an
ffQ
equation of the form
= MQ,
where isany function of u and w, determine a surface S upon which the parametric curves are the asymptotic lines, and every
other solution linearly independent of these three gives by ratures an infinitesimal deformation of S.
M
quad
EXAMPLES
1.
A
tion (2) be applicable
2.
ft,
necessary and sufficient condition that two surfaces satisfying the condi is that they be minimal surfaces adjoint to one another.
If x, y, z
and x l9
T/I,
zi satisfy the
condition
(2),
so also
do
,
17,
f and &,
^j,
the latter being given
by
77
= aix + =2+ = a 3 x f f
,
biy
&22/
68 y
,
+ + +
ciz
C2 Z
c3z
+ + +
di,
xi
2/1
d2
,
d8
,
Zi
= a^ + a2 Tn + a s ft + ei, = &ll + &2^?l + &3ft + C = Ci^ + c 2 + c 3 ft + c 8
2
,
&gt;?i
,
where a 1? a 2
3.
,
ei, e 2
e3
are constants.
necessary condition that the locus of the point (xi, ?/i, z\) be a curve is that S be a developable surface. In this case any orthogonal trajectory of the tangent planes to S satisfies the condition.
4.
5.
A
Investigate the cases
If Si
=
and
to
= c, where c is a constant different from zero.
of linear elements, so also does the locus of a point dividing in constant ratio the line joining corresponding points on Si and S{.
and S{ correspond
S with orthogonality
378
INFINITESIMAL DEFORMATION
The expressions
in the parentheses of
155. Associate surfaces.
equation
efficients,
(10) differ
D
,
Z&gt;
Z&gt;J,
",
only in sign from the second fundamental co of the surface /7 enveloped by the plane
(14)
Hence equation
(15)
(10)
may
D"D
be written
Q
+ DDJ  2 D D[ = 0.
the condition that to the asymptotic lines upon either of the surfaces S, S there corresponds a conjugate system on
is
( 56). Bianchi applies the term associate to two sur whose tangent planes at corresponding points are parallel, and for which the asymptotic lines on either correspond to a conjugate system on the other. Since the converse of the pre
This
the other
faces
ceding results are readily shown to be true, of Bianchi f
:
we have
the theorem
When
from a
two surfaces are associate the expression for the distance fixed point in space to the tangent plane to one is the char
acteristic
function for an infinitesimal deformation of the other.
the problems of infinitesimal deformation and of the determination of surfaces associate to a given one are equivalent. consider the latter problem.
Hence
We
Since the tangent planes to
are parallel,
S and SQ
dxn 5
dv
2
at corresponding points
we have
dzn
2
dx = X 
dx
fji
,
=
0
dx  r dx
du
du
dv
du
X,
ft,
du
and similar equations in y Q and of u and v to be determined. J
,
where
by
&lt;r,
r are functions
If these equations be multiplied
~y ~Y
likewise by
dv
dv
2V
and added, we
dv
&M obtain
$U
 and added, and dU
\D&lt;
t
* Cf.
J
67.
Lezioni, Vol.
II, p. 9.
The negative
signs before
p.
and r are taken so that subsequent
results
may have a
suitable form.
ASSOCIATE SURFACES
where
Z&gt;
379
When
(17)
are the second fundamental quantities for $. these values are substituted in (15), we find
"
,
Z&gt;
,
D
Xr=0.
Consequently the above equations reduce to
du
If
du
of the
dv
dv
du
dv
we make use
Gauss equations (V,
is
7),
the condition of
integrability of equations (18)
reducible to
du
dv
where
A and B
2,
are determinate functions.
Since similar equations
Calculating the following equations
hold in y and
both
A
and
B must be identically zero.
we have
&lt;r
the expressions for these functions, to be satisfied by X, /A, and
:
JL
d\
.
f22i
.
rm
fl21
I1J
.
rii
(19)
da_d\
du
dv
J22\
v
fill
I
2
1J
1J
To
(20)
these equations
we must add
2
\D 
&lt;rZ&gt;
=
pl&gt;"
0,
obtained from the last of (16).
The determination
of the asso
ciate surfaces of a given surface referred to any parametric system requires the integration of this system of equations. Moreover,
shall now every set of solutions leads to an associate surface. consider several cases in which the parametric curves are of a
We
particular kind.
face
Suppose that S is a sur which the parametric curves form a conjugate system. upon We inquire under what conditions there exists an associate sur face upon which also the corresponding curves form a conjugate
156. Particular parametric curves.
system.
380
INFINITESIMAL DEFORMATION
this hypothesis
On
we
have, from (16),
/*
=
a
=
6,
so that equations (19) reduce to
&lt;&gt;
which are consistent only when
that
is,
when
the point equation of
S,
namely
j^
dudv
fi21&lt;tf
\l
J
cu
\ZJfo
ri2&lt;tf
has equal invariants
(cf.
165).
the function X Conversely, when condition (22) is satisfied, the equations makes given by the quadratures (21)
of an associate surface are compatible, and thus the coordinates have the theorem of Cosserat*: obtained by quadratures. Hence we
a surface S is the same problem infinitesimal deformation of as the determination of the conjugate systems with equal point invari
The
ants on S.
reciprocal and the both surfaces, these curves on parametric curves are conjugate for also have equal point invariants.
Since
the
relation
between S and S
is
lines asymptotic lines, the corresponding In this case, as is seen from (16), on SQ form a conjugate system. X is zero and equations (18) reduce to
If
S
be referred to
its
.
/24
&gt;
dxn
2
)
=
u
dx
dxn
dv
dx
a
.
;
du
^dv
3u
moreover, equations (19) become
n
Toulouse Annales, Vol. VII (1893), N.
60.
KULED SUEFACES
381
The solution of this system is the same problem as the integra tion of a partial differential equation of the second order, as is seen by the elimination of either unknown. When a solution of
the former
is
obtained,
the
corresponding value of the other
unknown is given directly by one of equations (25). We make an application of these results to a ruled surface, which we suppose to be referred to its asymptotic lines. If the
curves v
const, are the generators, they are geodesies,
and conse
quently (VI, 50)
p
1
12
can be found by a quadrature. When this value is sub stituted in the second of equations (25), we have a linear equa tion in and consequently also can be obtained by quadratures.
Now
/*
&lt;r
&lt;r,
Hence we have the theorem
:
When
the curved asymptotic lines
on a ruled surface are known,
its associate
surfaces can be
found by quadratures.
its
If
S were
referred to
(24).
asymptotic
lines,
we should have
equations similar to as follows:
These equations may be interpreted
The tangent
is
parallel to
on one of two associate surfaces the direction conjugate to the corresponding curve on
to
an asymptotic
line
the other surface.
EXAMPLES
1.
If
two associate surfaces are applicable
to
one another, they are minimal
surfaces.
2.
Every surface of translation admits an associate surface
of translation such
that the generatrices of the two surfaces constitute the
3.
4.
common
conjugate system.
The surfaces
associate to a sphere are minimal.
When
the equations of the right helicoid are
x
u cos v,
y
u
sin
u,
z
cro,
2 2 the characteristic function of any infinitesimal deformation is = ( V) (u + )~ functions of u and v respectively. Find the surfaces are arbitrary and where
U+
,
U
V
Si and So, and
5.
show that the
latter are
molding surfaces.
If
S and S are
associate surfaces of a surface S, the locus of a point
dividing in constant ratio the joins of corresponding points of So
associate of S.
and S6
is
an
382
INFINITESIMAL DEFOKMATION
S1? S
.
157. Relations between three surfaces S,
Having thus
discussed the various ways in which the problem of infinitesimal deformation may be attacked, we proceed to the consideration of
other properties which are possessed by a set of three surfaces
$,
Stf
SQ
.
We
recall the differential equation
dxdxl + dydy
v
+
dzdz l =
0,
and remark that
(26)
if
it
may
yQ dz,
,
be replaced by the three
dXi=zQ dy
dy^ x dz
Q
z Q dx,
dz^y^dx
x^dy,
the functions #
?/
,
z
are such a
form that the conditions of
integrability of equations (26) are satisfied.
These conditions are
du dv
du dv
dx_d_z_o
dv du
dv du
d_x_dz,
= fe?5,
dv du dv du
du dv
dx dy,
du dv
dv du
du dv
If these equations
du dv
dy dx dv du
(
be multiplied by
i
"


respectively and
F, ^,
added, and likewise by
?
and by
JT,
we
obtain,
0y
"
by (IV,
(27)
2),
(28)
From the first two of these equations it follows that the locus of the point with coordinates XQJ T/O z corresponds to S with paral lelism of tangent planes.
,
from
In order to interpret the last of these equations we recall 61 that
a d(Y, Z}
ft d(u,
v)
Y=
a d(Z,
ft d(u,
X)
v)
a d(X, Y)
/if
d (u, v)
KELATIONS BETWEEN
where a
is
S,
S
19
AND
S
383
or negative.
1 according as the curvature of the surface is positive If we substitute these values in the lefthand
mem
bers of the following equations, and add and subtract dU dv dx dX dX from these equations respectively, the resulting and
members
dv du dv
expressions are reducible to the form of the righthand
du/
\
du
dv
dv
\
du
dv
(28) can be
By means
of these
and similar
identities, equation
transformed into
,_
^. __ D,^
is
,
*
Y
o
Since this equation
quantities # when a surface
,
a;
ZQ
is
in
equivalent to (15) because of (27), the Hence (26) are the coordinates of S
.
ing surface S and Sr This result enables us to find another property of If X^ Y^ Zt denote the directioncosines of the normal to S^
they are given by
1
l I l
Sl known, the coordinates of the correspond are readily found.
d(u, v)
HI
F*,
d(u, v)
H^
d(u,
v)
where 7/
=
t
^^
E^ F^ G being
l
the coefficients of the
,
,
linear element of
values of the derivatives of x v y v z x r as given by (26), be substituted in these expressions, we have,
If the
in consequence of (14),
(30)
X^^
normal
to
Y
the theorem
:
As an immediate consequence we have
A
S
t
is
parallel
to the
radius vector of
S
at the corre
sponding point.
384
INFINITESIMAL DEFORMATION
of (30) we find readily the expressions for the second coefficients J9 t D[, If we notice that J of $L
,
By means
fundamental
D
.
and substitute the values from
1
(6)
and
(30) in
du du
^ du
dv
^4
dv
du
we
obtain
(31)
From
(32)
these expressions follow
Combining
S,
or,
this result
lines
to
with
(15),
we have
:
The asymptotic
Sv SQ
correspond
upon any one of a group of three surfaces a conjugate system on the other two;
in other
words:
lines
The system of
f
which
to the
is
conjugate for any two of three surfaces
*S ,
Sv S
If the
corresponds
asymptotic lines on the other.
negative,
its
curvature of
S be
asymptotic lines are
1
real,
is
and consequently the common conjugate system on S and S
real.
If these lines
be parametric, the second of equations (32)
reduces to
As an odd number of the four quantities in this equation must be negative, either SQ or S1 has positive curvature and the other negative. Similar results follow if we begin with the assumption that Sl or SQ has negative curvature.
If the
curvature of
l
S be
positive, the conjugate
system
common
to
it
and S
is
real
(cf.
56)
;
consequently the asymptotic lines
RELATIONS BETWEEN
on SQ are
real,
S,
S
AND
S
385
and the curvature
the curvature of
is
of the latter
we saw
that of
that
when
SQ
:
is
is negative. But negative, and of S positive,
S
1
also
negative.
Hence
S,
Given a set of three surfaces has positive curvature.
S^ SQ ; one and only one of them
Suppose that S
is
to asymptotic lines
on SQ
referred to the conjugate system corresponding The point equation of S is
.
We
If
shall prove that this
is
the point equation of S^ also.
we
differentiate the equation
with respect to v, and make use of the fact that tions of (33), we have, in consequence of (26),
~~
y and
z are
solu
dudv
\dv du~~l)vduj
I1
/ #M
t
2 J~dv
is zero in consequence of equa and hence xl is a solution of (33). Since the parametric curves on S are its asymptotic lines, the and consequently of S must satisfy spherical representation of
But
the expression in parenthesis
tions similar to (24),
the condition
^
f!2V
d
fl2V
Hence we have the theorem
The problem of
invariants
of Cosserat:
infinitesimal deformation of a surface is the
same
as the determination of the conjugate systems with equal tangential
upon
the surface.
158. Surfaces resulting
pass to tesimal deformation of 8.
from an infinitesimal deformation. We the consideration of the surface S arising from an infini
Its coordinates are
given by
where
a small constant whose powers higher than the first are for neglected. Since the fundamental quantities of the first order $ namely G are equal to the corresponding ones for by
is
,
J"
,
F
f
,
,
,
386
INFINITESIMAL DEFORMATION
means
of (26) the expressions for the directioncosines of the normal to S are reducible to
X
1
,
FZ
,
and similar expressions
for
Y
and
Z
.
The
means
derivatives of
of (29) to
X
/
with respect to u and v are reducible by
dX^_d_X
dY_
OJL
77
dZ\
ea /
D ,^_ D 3X
_
T)
.1
dX
dv
dX
i
fly
\
I
vZ\
I
I
d

I
(
uX
7)
i
vJL
dv
dv
dv /
/i
\
du
dv
where a
When
1 according as the curvature of $ is positive or negative. is these results are combined with (26) and (34), we obtain
^^_V ^4
du du
e
du du
/if
V du (D ^\
**l \9* 9*
The
+du
du cu)
D
dv
v
*\du
du
du du
it
last expression is identically zero, as
one sees by writing
for
out in full. a aTl
From
this
,
and similar expressions
,
X
,
and
dv
du
V dv **
,
v/
dv
VA
fix
1
flX 9
ay
^
the values for the second fundamental
in the form
coefficients of
S can be given
=(30)
2) u T** cu
T
=D + jf
We know
of
that ff
is
equal to
UK according
as the curvature
157, one and only 60). Also, by positive or negative (cf. surfaces S, Sv S has positive curvature. Recalling one of three 1 according as the curvature of that a in the above formulas is S is positive or negative, we can, in consequence of (31), write equations (36) in the form
S
is
Z&gt;
.*"
= .ZX
J&gt;,
where the upper sign holds when Sl has positive curvature.
ISOTHERMIC SUBFACES
From
these equations
it is
387
seen that
zero.
&
and
D
r
can be zero sim
ultaneously only when D[
is
Hence we have:
infini
The unique conjugate system which remains conjugate in an
tesimal deformation of a surface
is
the one corresponding to a conju thing, to the asymptotic lines
gate system on
S^, or,
what
is the
same
on
SQ
.
In particular, in order that the curves of this conjugate system
be the lines of curvature,
it
is
spherical representation be orthogonal, a minimal surface (cf. 55). From this
necessary and sufficient that the and consequently that be
it
representation of the lines of curvature of
versely,
if
follows that the spherical S is isothermal. Con
unique minimal sur same representation of its asymptotic lines, and this surface can be found by quadratures. Hence the required infinites
a surface
is
of this kind, there is a
face with the
imal deformation of the given surface can be effected by quadra tures (26), and so we have the theorem of Weingarten *
:
sufficient condition that a surface admit an infini tesimal deformation which preserves its lines of curvature is that the
A
necessary and
spherical representation of the latter be isothermal; when such a surface is expressed in terms of parameters referring to its lines of
curvature, the deformation can be effected by quadratures.
159. Isothermic surfaces.
By means
of the results of
158 we
obtain an important theorem concerning surfaces whose lines of curvature form an isothermal system. They are called isothermic
surfaces
(cf.
Exs.
1, 3, p. 159).
it
From
equations (23)
follows that
if
the
common
conjugate
system on two associate surfaces is orthogonal for one it is the same for the other. In this case equation (22) reduces to
of
which the general integral
is
E
where
U
Hence the
41).
G=r
Akademie zu Berlin,
1886.
functions of u and v respectively. lines of curvature on S form an isothermal system (cf.
*
U and V are
Sitzungsberichte der Konig.
388
If the
INFINITESIMAL DEFORMATION
parameters be isothermic and the linear element written
ds 2 =r(du
2
+dv
2
),
it
follows from (21) that
(37)
X
(23)
= i,
~
and equations
become
~~
du
r du
dv
r dv
From
these results
:
we
derive the following theorem of
Bour * and
to its lines
Christoffel
If the linear element of an isothermic surface referred
of curvature be
ds
*
_r
/
du
*
__
dl?\
It is asso
a second isothermic surface can be found by quadratures. ciate to the given one, and its linear element is
1
ds?
r
(du
+ dv
).
From equations (16) and (17) it follows that the equation of the common conjugate system (IV, 43) on two associate surfaces $, S
is
reducible to
fi
(38)
du 2 + 2 X dudv
+ a dv = 0.
2
tion that
The preceding results tell us that a necessary and sufficient condi S be an isothermic surface is that there be a set of solu
is
tions of equations (19) such that (38)
of curvature
on
S.
Hence there must
2
the equation of the lines be a function p such that
&lt;r
p
p (ED*
FD),
X
==
p
(ED
1
GD),
=?&lt;p
(FD"
GD
f
)
satisfy equations (19).f equations of the form
Upon
Ss
substitution
we
are brought to
two
=
a;,
= p\
u and
v.
du
dv
where a and
ft
are determinate functions of
In order that
S be
isothermic, these functions
must
~~
satisfy the condition
dv
du
When
it is satisfied,
p and consequently p, X, a are given by quad
/
ratures.
* Journal de
f Cf.
I Ecole Poly technique, Cahier 39 (18G2), p. 118. Bianchi, Vol. II, p. 30.
ISOTHEKMIC SURFACES
Consider furthermore the form
(39)
389
H(p du* +
2
X dudv
+
o
dv~).
lines of curvature are para (37) it is seen that when the Hence its curvature is to 2 dudv. metric, this expression reduces
From
zero
(cf.
From
ratures.
V, 135
12),
it
follows that this form
and consequently the curvature of (39) is zero. is reducible to du 1 dv l by quad
the theorem of Weingarten
:
Hence we have
The
lines of curvature
upon an isothermic surface can
be
found by
quadratures.
We
of Ribaucour.
conclude this discussion of isothermic surfaces with the proof of a theorem He introduced the term limit surfaces of a group of applicable sur
faces to designate the members of the group whose or minimum. According to Ribaucour,
mean curvature
is
a
maximum
The limit surfaces of a group of applicable surfaces are isothermic.
In proving
vature.
(36) the
Its
it we consider a member S of the group referred to its lines of cur mean curvature is given by D/E + D"/G. In consequence of equations mean curvature of a nearby surface is, to within terms of higher order,
A
or
necessary and sufficient condition that the
mean curvature
of
S be a maximum
minimum
is
consequently
/j)
j&gt;"\
Excluding the case of the sphere for which the expression in parenthesis we have that DO is zero. Hence the common conjugate system of S and posed of lines of curvature on the former, and therefore S is isothermic.
&lt;S
is
zero,
is
com
GENERAL EXAMPLES
the coordinates of two surfaces corresponding with surfaces orthogonality of linear elements, the coordinates of a pair of applicable are given by =+ = y + ty\, =x+ n fc
1.
If x, y,
and
xi, y\, z\ are
tei,
txi,
m
2
&lt;zi,
2
=x
r}
y
tyi,
f2
=
z
tei,
where
2.
any constant. two surfaces are applicable, the locus of the midpoint of the corresponding points admits of an infinitesimal deformation in which
t is
If
line joining
this line is
the generatrix.
3.
tion
(a, 6,
Whatever be the surface S, the characteristic equation (7) admits the solu = aX + bY + cZ, where a, 6, c are constants. Show that S is the point c) and that equations (26) become
infinitesi
 bz + d, ex \ e, ay + /, z\ = bx yi = az xi = cy where d, e, /are constants; that consequently Si is a plane, and that the mal deformation is in reality an infinitesimal displacement.
390
4.
INFINITESIMAL DEFORMATION
Determine the form of the
results of Exs.
1, 2,
where
has the value of Ex.
3.
5.
Show
that the
first
fundamental
coefficients EI, FI,
GI of a surface Si are of
the
form
E=
1
E&lt;f&gt;*
,
,
= F0 2
_.
dv
Let S denote the locus of the point which bisects the segment of the normal S between the centers of principal curvature of the latter. In order that the lines on 2 corresponding to the lines of curvature on S shall form a conju
6.
to
a surface
it is necessary and sufficient that S correspond to a minimal surface with orthogonality of linear elements, and that the latter surface and S correspond with parallelism of tangent planes.
gate system,
7.
Show
that
when
face
S
satisfies the
the spherical representation of the asymptotic lines of a sur condition a (92 \\y
cu
(
2
}
cv
(
1
= and equations (25) admit two pairs of solutions which are such that /x = On the two associate surfaces S SQ thus found by quadratures the parametric systems are isothermalconjugate, and S Q and S Q are associates of one another.
&lt;r
/j.
&lt;r.
,
8.
Show
two surfaces associate
9.
that the equation of Ex. 7 is a necessary and sufficient condition that to S be associate to one another.
Show
that
when
the sphere
is
referred to
its
minimal
lines, the
condition of
Ex. 7
is satisfied,
and investigate
this case.
10. On any surface associate to a pseudospherical surface the curves correspond ing to the asymptotic lines of the latter are geodesies. A surface with a conjugate system of geodesies is called a surface of Voss (cf. 170).
11.
Determine whether minimal surfaces and the surfaces associate to pseudo
spherical surfaces are the only surfaces of Voss.
12.
When
the equations of a central quadric are in the form (VII, 35), the asso
ciate surfaces are given
by
2/o
= 2 V6
Fj Uu du + f Vv dv\
u and
,
z
=i
v respectively
;
where
13.
and are arbitrary functions of are surfaces of translation.
U
V
hence the associates
When
the equations of a paraboloid are in the form
x=Va(u +
1&gt;),
y=Vb(uv),
z
=
2uv,
;
the associate surfaces are surfaces of translation whose generators are plane curves their equations are
x
= Va(U + V),
y
=Vb(VU),
z
= 2fuU
du
where
U and V are
arbitrary functions of u and v respectively.
GENERAL EXAMPLES
14.
391
Show
its lines
of curvature,
that a quadric admits of an infinitesimal deformation which preserves and determine the corresponding associate surface.
&lt;S
between S arid Si is reciprocal, there is a surface 3 which bears to S a relation similar to that of SQ to Si. Show that the asymptotic lines on S and S 3 correspond, and that these surfaces are polar 2 2 + z 2 + 1 = 0. reciprocal with respect to the imaginary sphere z +
15. Since the relation
associate to Si
?/
reciprocal, there is a surface S% cor responding to S with orthogonality of linear elements which bears to S a relation
is
16. Since the relation
between S and So
similar to that of Si to So. Show that the asymptotic lines on Si and that the coordinates of the latter are such that
Sz
correspond,
xixz =
and that the
17.
yzo

zy
,
yiy z =
zx

zz
,
z\
is
z*
=
xy

yx
,
line joining
corresponding points on Si and S 2
tangent to both surfaces.
Show
that
of linear elements
S& are related to
18.
S 5 denotes the surface corresponding to S 3 with orthogonality which is determined by Si, associate to SB, the surfaces S and one another in a manner similar to Si and Sz of Ex. 16.
if
Show
that the surface
&lt;S
the polar reciprocal of
19. If
S 4 which is the associate to Sz determined by So, with respect to the imaginary sphere x 2 + y 2 + z 2 + 1 =
,
is
0.
we continue
the process introduced in the foregoing examples,
we
obtain
two sequences
of surfaces
S,
Si, So,
$3,
Sj,
S7
,
Sg, $8,
Sn,
Sio,

,
S,
S2
,
84,
Se,
Show
that
Sn and S 10
are the same surface, likewise
;
quently there is a closed system of twelve surfaces faces of Darboux.
20.
lines
S i2 and S 9 and that conse they are called the twelve sur
,
A necessary and sufficient condition that a surface referred to be isothermic is that j) jj
D"
its
minimal
F
iso
where
21. of curvature on an thermic surface be represented on the sphere by an isothermal system is that
U and V are functions of u and v respectively. A necessary and sufficient condition that the lines
P*
Pi_U ~ F
where
and are functions of u and v respectively, the latter being parameters referring to the lines of curvature. Show that the parameters of the asymptotic lines on such a surface can be so chosen that = G.
U
V
E
22.
Show
that an isothermic surface
is
transformed by an inversion into an
isothermic surface.
23. If Si and S 2 are the sheets of the envelope of a family of spheres of two parameters, which are not orthogonal to a fixed sphere, and the points of contact of any sphere are said to correspond, in order that the correspondence be conformal,
it is
necessary that the lines of curvature on Si and surfaces be isothermic (cf. Ex. 15, Chap. XIII).
S 2 correspond and that
these
CHAPTER
XII
RECTILINEAR CONGRUENCES
160. Definition of a congruence. Spherical representation.
A two
parameter system of straight lines in space is called a rectilinear congruence. The normals to a surface constitute such a system ;
likewise the generatrices of an infinitesimal deformation of a sur face (cf. 152). Later we shall find that in general the lines of a congruence are not normal to a surface. Hence congruences of
normals form a special class
;
they are called normal congruences.
of light.
They were
the
first
studied, particularly in investigations of the
effects of reflection
and refraction upon rays
The
first
purely mathematical treatment of general rectilinear congruences was given by Kummer in his memoir, Allgemeine Theorie der gradlinigen Strahlensysteme.* We begin our treatment of
the subject with the derivation of certain of methods similar to his own.
Rummer s
results
by
the definition of a congruence it follows that its lines meet a given plane in such a way that through a point of the plane one line, or at most a finite number, pass. Similar results hold if a surface be taken instead of a plane this surface is
;
From
called the
surface
of reference.
And
so
we
.
r
rnay define a con
gruence analytically by means of the coordinates of the latter surface in terms of two parameters u, v, and by the directioncosines of the lines in terms of these parameters. Thus, a con
gruence
is
defined by a set of equations such as
*f\( u
where the functions
i
v}
&gt;
y
= fz( u
v
&gt;
z
)&gt;
jz( u v )
&gt;
&gt;
/ and
&lt;
under consideration, and the functions
are analytic in the domain of are such that
(/&gt;
u and
v
* Crelle, Vol.
LVII
(1860), pp. 189230.
302
NORMAL CONGRUENCES
393
a representation of the congruence upon the unit sphere by drawing radii parallel to the lines of the congruence, and call it the spherical representation of the congruence. When We put
We
make
the linear element of the spherical representation
(3)
is
da
2
=
2
&lt;f(^
+ 2 &dudv
dX
dx
If
we put
= ^A
we have
(5)
dx
dX
,,
ex
dX
dx
the second quadratic form
]
dxdX= e du + (/ +/
2
)
rfwdv
+ g dv\
If
which
fundamental in the theory of congruences. 161. Normal congruences. Ruled surfaces of a congruence.
is
there be a surface
of
(6)
S normal
to the congruence, the coordinates
S
are given
by
y
x
t
=x + tX,
=y + tY,
z
=z + tZ,
r
.
where
Since
(7)
measures the distance from the surface of reference to S is normal to the congruence, we must have
which
is
equivalent to
du
du
dv
3v
If these equations be differentiated
tively,
(9)
with respect to v and u respec and the resulting equations be subtracted, we obtain
/=/
.
Conversely, when this condition is satisfied, the function t given by the quadratures (8) satisfies equation (7). Since t involves an
additive constant, equations (6) define a family of parallel surfaces normal to the congruence. Hence
:
A
that
necessary and sufficient condition for a normal congruence be equal. f and
is
f
the congruence which pass through a curve on surface of reference S form a ruled surface. Such a curve, and
The
lines of the
394
RECTILINEAR CONGRUENCES
is
consequently a ruled surface of the congruence,
relation
(10)
determined by a
between u and
v.
Hence
a differential equation of the form
Mdu+Ndv =
consider defines a family of ruled surfaces of the congruence. a line l(u, v) of the congruence and the ruled surface 2 of this family upon which I is a generator ; we say that 2 passes through I. 103, 104. apply to 2 the results of
We
We
If dsQ
denotes the linear element of the curve
it
C
in
follows from (VII, 54), 2 the quantities a and b for 2 have the values
the surface of reference,
which 2 cuts (3), and (5) that
\*
da 2
~ ^ dX
From common
v
dx
(VII, 58)
we have
I
that the directioncosi ies X,
//.,
v of the
4 du,
perpendicular to
is
and
to the line
(10),
+
dv,
where dv/du
given by
I of parameters u have the values
(12)

\
da
da
which, by means of (V, 31), are reducible to
dX  ^dX\ &
(13)
,
\=
dv
)du du /
/ + ^dX  3 dv
,
*&gt;dX\,
{
&lt;?
dv
\
du/
&* da
.
and similar expressions
for
/A
and
v.
From
(12)
it
follows that
.dX X
^~ + da
,
dY
^^~"
da
hz/
t
dZ n ^~ =0 da

Since dX/da, dY/da, dZ/da are the directioncosines of the tangent to the spherical representation of the generators of 2, we have the
theorem
:
Given a ruled surface 2 of a congruence ; let C be the curve on the point of C correspond the unit sphere which represents 2, and
M
ing
to
a generator
to
L
of
S;
the limiting position of the
common per
to the
pendicular
tangent
to
L and
M.
a nearby generator of
2
is
perpendicular
C
at
PRINCIPAL SURFACES
395
162. Limit points. Principal surfaces. By means of (VII, 62) and (12) we find that the expression for the shortest distance 8 between I and V is, to within terms of higher order,
dx
dsf
da
dy
dz
X
Y
dS
z
dZ
dX dY
When
member
(14)
the values (13) for X, /*, v are substituted in the righthand of this equation, the result is reducible to
&lt;odu
/{do
e
du
du
+ gdv + g dv
,
If jY denotes the point where this line of shortest distance meets the locus of jVis the line of striction of 2. Hence the distance of
N from
it
the surface
r,
be denoted by
.
measured along Z, we have, from (11),
$,
jCJ.2
is
given by (VII, 65)
;
if
n/ 10
V
r
2 _ edu*+(f+f) dudv + g dv
i
For the present* we exclude the case where the
coefficients of
the two quadratic forms are proportional. Hence r varies with the value of dv/du, that is, with the ruled surface 2 through I. If
we
limit our consideration to real surfaces 2, the denominator
is
always positive,
and consequently the quantity
r has a finite
mum
and minimum. In order
to find the surfaces
,
2
for
maxi which r
has these limiting values,
(16)
we
replace dv/du by
and obtain
If
we equate
,
to zero the derivative of the righthand
member with
respect to
we get
a quadratic in t. Since $&*&gt; 0, we may apply to this equation reasoning similar to that used in connection with equation (IV, 21),
* Cf. Ex.
1,
171.
396
RECTILINEAR CONGRUENCES
and thus prove that it has two real roots. The corresponding values of r follow from (16) when these values of t are substituted in the latter. Because of (17) the resulting equation may be written
r
where
t
When we
indicates a root of (17) and r the corresponding value of write the preceding equations in the form
r.
r
\
+
e]
+
[$r
+\
o,
and eliminate
t,
we
obtain the following quadratic in r:
If r^
and
r 2 denote the roots of this equation,
we have
(19)
The
which
points on
limit points.
lie
corresponding to these values of r are called its They are the boundaries of the segment of I upon
I
nearby line of the congruence.
the feet of each perpendicular common to it and to a The ruled surfaces of the con,
I
gruences which pass through
and are determined by equation
(17) are called the principal surfaces for the line. There are two of them, and their tangent planes at the limit points are determined
by
I
and by the perpendiculars
of shortest distance at the limit
points.
They
are called the principal planes.
In order to find other properties of the principal surfaces, we imagine that the parametric curves upon the sphere represent these surfaces. If equation (17) be written
du
(20)
iu
+
=
l
0,
PRINCIPAL SURFACES
it is
397
surfaces v
seen that a necessary and sufficient condition that the ruled = const., u = const, be the principal surfaces, is
From
thes"e
it
follows that since the coefficients of the two funda
mental quadratic forms are not proportional, we must have
(21)
^=0,
the
first
/+/
=0.
From
of these equations
and the preceding theorem follows
the result:
The principal surfaces of a congruence are represented on the sphere by an orthogonal system, and the two principal planes for
each line are perpendicular
to
one another.
For
this particular parametric
system equation (13) reduces to
&lt;o
^9X, du
^dX. dv &
(22)
so that the directioncosines
is
X
x,
JJL^
v l of the
the limit point on
I
corresponding to v
1
= const,
perpendicular whose foot have the values
l
ay
dz
Hence the angle
and those with
GO
between the
is
lines
with these directioncosines
.
(22)
given by
cos
&&gt;
=
7 du
The values
of r 1
and r 2 are now
e
ri=r
r
^~S ^
2
a
so that with the aid of (23) equation (15) can be put in the form
(24)
r
is
=r
l
cos
2
&&gt;
f r 2 sin
co.
This
Hamilton
s
equation.
We
remark that
it is
independent of
the choice of parameters.
398
KECTILINEAR CONGEUENCES
163. Developable surfaces of a congruence. Focal surfaces. In order that a ruled surface be developable, it is necessary and suffi cient that the perpendicular distance between very near generators
be of the second or higher order. From (14) it follows that the ruled surfaces of a congruence satisfying the condition
(25)
e
du
}fdv,
f
du
+ g dv
are developable.
fying this equation are
Unlike equation (20), the values of dv/du satis not necessarily real. We have then the
theorem
:
Of all
the ruled surfaces of
a congruence through a line of
it
two
are developable, but they are not necessarily real.
The normals to a real surface afford an example of a congruence with real developables for, the normals along a line of curvature form a developable surface ( 51). Since /and/ are equal in this
;
case, equations (20)
and
(25) are equivalent.
And, conversely, they
are equivalent only in this case.
Hence
:
When
a congruence
is
normal, and only then, the principal surfaces
are developable.
When
them.
a ruled surface
is
developable
its
generators are tangent
to a curve at the points
where the
lines of shortest distance
is
meet
Hence each
line of a
congruence
tangent to two curves
in space, real or imaginary according to the character of the roots of equation (25). The points of contact are called the focal points
for the line.
By means
e
of (25)
we
find that the values of r for
these points are given by
du \fdv
_
f du+g dv
If these
equations be written in the form
(&lt;&!&gt;
(p 4 e)du + (&p +f)dv = 0, +/ du 4 (gp +g}dv = 0,
)
and
(26)
if
du, dv be eliminated,
we have
DEVELOPABLE SURFACES
If p l
399
and p 2 denote the roots of
this equation, it follows that
(27)
A= ^//
(19)
From
and
(27) it is seen that
(28)
These results may be interpreted
The midpoints of
limit points
as follows
:
the two segments
bounded respectively by
the
and by
the focal points coincide.
its
This point is called the middle point of the line and middle surface of the congruence.
locus the
The distance between the focal points
between the limit points.
is
never greater than that
the congruence is normal.
They
coincide
when
Equation
(24)
may
2
be written in the forms
cos
&)
=
ri
A*
T
1
r2
sin
2
&)
=
r i
&)
r
r
i
Hence
if a) 1
and
&)
2
denote the values of
corresponding to the
developable surfaces,
we have
^,
A*
From
these and the
cos
2
first
ft)
of (28)
2
it
follows that
sin
2
ft)
1
= sin
&)
2,
1
= cos
2
ft)
2,
so that
(29)
cos2&) 1 +cos2ft) 2 =0,
and consequently
(30)
jor
w 1 +o) 2
~ft) 2
=ww,
=
(31)
ft)
1
400
RECTILINEAR CONGRUENCES
where n denotes any integer.
If the latter equation be true, the developable surfaces are represented on the sphere by an orthog onal system, as follows from the theorem at the close of 161. But
by
tem on the sphere is/=/ that
,
34 the condition that equation (25) define an orthogonal sys is, the congruence must be normal.
Since in this case the principal surfaces are the developables, equa tion (30) as well as (31) is satisfied. Hence equation (30) is the
general solution of (29).
The planes through I which make the angles o^, = are called the focal planes for the principal plane w
are the tangent planes to the
&&gt;
2
with the
;
line
they
two developable surfaces through
the line.
Incidentally
we have proved the theorem:
that the congruence
A
necessary and sufficient condition that the two focal planes for
is
each line of a congruence be perpendicular
be normal.
And from
equation (30)
it
follows that
The focal planes are symmetrically placed with respect to the prin cipal planes in such a way that the angles formed by the two 2iairs
of planes have the same bisecting planes.
If 6
denote the angle between the focal planes, then
^
sin 6
and
(32)
cos 2
a)
l
cos
2
o)
1
cos
2
o&gt;
2
=
l
The
loci of the focal points of a
surfaces.
Each
line of the
congruence are called its focal congruence touches both surfaces, being
it.
tangent to the edges of regression of the two developables through
By reasoning
of center
(
similar to that
employed
in the discussion of surfaces
:
74)
we prove
the theorem
A
congruence
faces.
regarded as two families of developable sur Each focal surface is touched by the developables of one family
may
be
along their edges of regression and enveloped by those of the other
family along
the curves conjugate to these edges.
The preceding theorem shows that of
a line
I
one
is
tangent to the focal surface
the two focal planes through SL and the other is the
ASSOCIATE NORMAL CONGRUENCES
401
osculating plane of the edge of regression on /S\ to which I is tan gent similar results hold for Sz When the congruence is nor mal these planes are perpendicular, and consequently these edges
;
.
of regression are geodesies on true ( 76), we have:
S and Sz
l
.
Since the converse
is
necessary and sufficient condition that the tangents to a family of curves on a surface form a normal congruence is that the curves be
geodesies.
A
EXAMPLES
1.
If JT,
Y",
Z
are the directioncosines of the normal to a minimal surface at
the point (cc, T/, z), the line whose directioncosines are F, through the point (x, y, 0) generates a normal congruence.
2.
X, Z and which
passes
tact of a
Prove that the tangent planes to two confocal quadrics at the points of con common tangent are perpendicular, and consequently that the common tangents to two confocal quadrics form a normal congruence.
3.
Find the congruence of common tangents
x
2
to the paraboloids
+
y
2
=
2az,
x2
+ L L
y*
=
2 az,
and determine the
4.
focal surfaces.
line
If
two ruled surfaces through a
lines, their lines of striction
orthogonal
meet
are represented on the sphere by at points equally distant from the
middle point.
5.
same angle,
In order that the focal planes for each line of a congruence meet under the it is necessary and sufficient that the osculating planes of the edges of
under regression of the developables meet the tangent planes to the focal surfaces
constant angle.
6.
A
ence be
its
necessary and sufficient condition that a surface of reference of a congru middle surface is g  (/ + )&lt;^+ e& = 0.
/
164. Associate normal congruences.
If
we put
dx
dx
equations
(34)
(8)
may
be replaced by
t
=c
Now
I
7 du
+
y^dv,
where
c is a
constant.
equation
(9) is
equivalent to
402
RECTILINEAR CONGRUENCES
may
be written
In consequence of this condition equation (34)
(36)
t
= c$(u^
orthogonal be taken as parametric curves
vl
If the
where u
vl
l
is
a function of u and v thus denned.
trajectories of the curves
=
u^ const,
const., it follows
from (36) and from equations in u l and
(34) that
analogous to (33)
and
From
The
this result follows the
lines of
theorem
:
a normal congruence cut orthogonally the curves on
t
the surface of reference at whose points
is
constant.
If denotes the angle which a line of the congruence makes with the normal to the surface of reference at the point of inter
section,
we have
sin
(37)
*=
is
where the linear element of the surface
If
S be taken
for the surface of reference of a second congruence
whose directioncosines
Xv
Yv Z
l
satisfy the conditions
u i) where normal and
4&gt;i(
ig
anv function whatever of u^
has the value
this
congruence
is
any function, there is a family of these normal congru ences which we call the associates of the given congruence and of
Since
1
is
one another.
Through any point
of the surface of reference there
lie
passes a line of each congruence, and all of these lines const, through the point. plane normal to the curve u l
in the
:
Hence
The two lines of two associate congruences through the same point of the surface of reference lie in a plane normal to the surface.
DERIVED CONGRUENCES
Combining with equation
gruence,
(38)
(37) a similar
403
one for an associate con
we have
E* =
sin^
:
#K)
&) =/(W)
&lt;*&gt;
Hence we have the theorem
The
congruences make with
ratio of the sines of the angles
which the
lines of two associate
is
the
normal
to their
t
surface of reference
constant.
con
stant along the curves at whose points
is
When in particular f(u^ in (38)
and equation
is
a constant, the former theorem
(38) constitute the laws of reflection
and refraction of
rays of light, according as the constant is equal to or different from minus one. And so we have the theorem of Malus and Dupin
:
If a bundle of rays of light forming a normal congruence be reflected
or refracted
any number of times by
the surfaces of successive
homo
geneous media, the rays continue
to constitute
a normal congruence.
By means
of (37) equation (36) can be put in the
t
form
=
c
l
\l
E sin 6 du r
:
From
a
this result follows the
theorem of Beltrami *
be
If a surface of reference of a normal congruence
deformed in such
that the directions of the lines of the congruence with respect to the surface be unaltered, the congruence continues to be normal.
way
165. Derived congruences. It is evident that the tangents to the curves of any oneparameter family upon a surface S constitute a congruence. If these curves be taken for the parametric lines
v
and their conjugates for u = const., the developables in one family have the curves v = const, for edges of regression, and
= const.,
u
the developables of the other family envelop S along the curves const. may take S for the surface of reference. If Sl be
We
the other focal surface, the lines of the congruence are tangent to the curves u = const, on Sr The tangents to the curves v = const,
S1 form a second congruence of which Sl is one focal surface, and the second surface $ 2 is uniquely determined. Moreover, the
on
* Giornale di matematiche, Vol. II (1864), p. 281.
404
lines of the
.
RECTILINEAR CONGRUENCES
second congruence are tangent to the curves u = const. on Sz In turn we may construct a third congruence of tangents This process may be continued const, on Sz to the curves v indefinitely unless one of these focal surfaces reduces to a curve,
.
or
is
infinitely distant.
In like manner
the curves
we get
u = const, on
on S_ l form
a congruence by drawing tangents to S, which is one focal surface, and the
other, S_ v is completely determined.
The tangents
to the curves
u
const,
still
another, and so on.
In this
way we
obtain a suite of surfaces
terminated only when a surface reduces to a curve, or points are infinitely distant. Upon each of these surfaces the parametric curves form a conjugate system. The congruences thus
which
is
its
obtained have been called derived congruences by Darboux.* It is clear that the problem of finding all the derived congruences of a given one reduces to the integration of the equation of its developables (25);
for,
when
its
the developables are
focal surfaces.
known we have
the
conjugate system on
In order to derive the analytical expressions for these results,
we
recall ( 80) that the coordinates x, y, z of of an equation of the form
S
are solutions
(39)
du dv
b are
du
cv
v.
where a and
nates of
l
determinate functions of u and
2t,
If the coordi
S be denoted by x^ y^
dx
they are given by
*x+\*
fcjr
By + x,.
v
dz.
+ Xi S
,
,
dz
.
where \^J~E measures the distance between the focal points. But as the lines of the congruence are tangent to the curves u = const. on Sv we must have
dx.
(40)
1
dv
= Pl dx M
1
dy. ^I
du
dv
* Vol.
= Pl dy u
1
*
i
=u
dz
l
,
du
dv
du
II,
pp. 1622.
DERIVED CONGRUENCES
where
is
/^
405
is
a determinate function of
is
u and
v.
When
the above
value for x l
substituted in the
first
of these equations, the result
reducible, by
means
L^LI
of (39), to
_ flX
dv
n\te + (1 __ fog l/
to
dv
V du
=
0.
Since the same equation is true for y and theses must be zero, that is,
1
a1
z,
the quantities in paren
a
Hence the surface S
,
l
is
defined by
1 ex
\dy
1 dz
,
and equations
/42\
i
(40)
become
__
__
^y
Vav b
b]du
dv
\dv b
b/du
dv
\dv b
is
bjdu
defined by
Proceeding in a similar manner, we find that $_i
the equations
/4 ox

1 a*

1 to
.
.
1
V*;

*
and that
and similar expressions in y_^ and 2_i. From (41) and (43) it is seen that the surface Sl or S.i a and 5 are both infinity, according as b or a is zero. When
is at
zero,
S
is
a surface of translation
generators of a surface of each of which the other focal surface
( 81). Hence the tangents to the translation form two congruences for
is
at infinity.
In order that S^ be a curve, x^ y^ z l must be functions of u alone. From (42) it follows that the condition for this is
d1
~dv
_a
~6*
is
5
In like manner the condition that $_i be a curve
l=i
du a
a
406
RECTILINEAR CONGRUENCES
&,
The functions h and
denned by
,
if
h __da du
*
dv
are called the invariants of the differential equation (39). the above results may be stated
:
Hence
A
_i be
necessary and sufficient condition that the focal surface Sl or a curve is that the invariant k or h respectively of the point
equation of
S
be zero.
166. Fundamental equations of condition. have seen ( 160) that with every congruence there are associated two quadratic dif ferential forms. Now we shall investigate under what conditions two quadratic forms determine a congruence. assume that we
We
We
a corresponding congruence. The tangents to the parametric curves on the surface of reference at a point are determined by the angles which they make with the
is
have two such forms and that there
tangents to the parametric curves of the spherical representation of the congruence at the corresponding point, and with the normal to
the unit sphere.
Hence we have the
relations
,__._+
I
Zti
7)ti
(44)
and similar equations
of
in
y and 2, where
#,
/3,
7; a r
(S v y l
r
are functions
respec
u and
v.
If
we multiply
also
these equations by
*
&gt;
tively,
and add;
dX dY dZ
by
dv dv dv
du
Y,
dU
3U
and by
A",
Z\ we obtain
from which we derive
a
/\
= e&
f
\
j c/ ^2 * (O& ~~ cy
j p= &
(n
iy
~~~
cc/
*
7
c/
=^.A 01^
&gt;
(45)
foo/
Ctf
CA
M*
fl
(Q^/
CV^
*&gt;2
c/
1
yi^^rr
1
FUNDAMENTAL EQUATIONS
In order that equations (44) be consistent, we must have
407
du \dv
2v \du
is
which, in consequence of equations (V, 22),
reducible to the form
]t?X+S X
du
dv
d
where
J?, S,
be satisfied
T are determinate functions. Since by Y and Z also, we must have R
a,
y3,
this equation
0,
must
S= 0, T= 0.
When
the values of
a^
fi v
from
(45), are substituted in these
equations,
we have
(47)
Conversely,
is
+1,
it
may
when we have a quadratic form whose curvature be taken as the linear element of the spherical rep
,
resentation of a congruence, which is determined by any set of functions e, f, / #, 7, 7^ satisfying equations (4T). For, when
these equations are satisfied, so also is (46), and consequently the coordinates of the surface of reference are given by the
quadratures (44).
Incidentally we remark that when the congruence is normal, and the surface of reference is one of the orthogonal surfaces, the last
of equations (47) is satisfied identically, and the first two reduce to the Codazzi equations (V, 27). apply these results to the determination of the congruences
We
with an assigned spherical representation of their principal surfaces, and those with a given representation of their developables.
167. Spherical representation of principal surfaces
and
of devel
opables.
A
necessary and
sufficient condition that the principal
surfaces of a congruence cut the surface of reference in the para metric lines is given by (21).
408
If
RECTILINEAR CONGRUENCES
we
require that the surface of reference be the middle surface of the congruence, and if r denote half the distance between the
limit points,
(48)
we
have, from (15),
e
r&lt;o,
g
= r&
first
When
these values are substituted in (47), the
two become
12,
(49)
f?
a/ /
\
7
=
^
1
d
i
,
(r
&lt;
and the
,50) 2
last is reducible to *
av
i
al
g^ g?
dv
\
dwy
cu
aiog^gTjg iog^,.
du
dv
dudv
+
YL a F W d { f l\42f * + sLNtfsfe;J 4N?s\^Jr^
d
\$
d
i
f
Moreover, equations (44) become
ao:
ax
/ ax
aa:
,
/ ax
ax
in
where 7 and y l are given by
(49)
;
and similar equations
y and
z.
reduces, therefore, to the determination of pairs of functions r and / which satisfy (50). Evidently either of these
Our problem
functions
may
be chosen arbitrarily and the other
is
found by the
solution of a partial differential equation of the second order. Hence any orthogonal system on the unit sphere serves for the
representation of the principal surfaces of a family of congruences, whose equations involve three arbitrary functions. In order that the parametric curves on the sphere represent the
developables of a congruence,
it is
necessary and sufficient that
from (25). If the surface of reference be the middle sur and p denotes half the distance between the focal points, it face, follows from (15) that e
as is seen
p
"
c
$
I,
* Cf. Bianchi, Vol.
p. 314.
DEVELOPABLES PARAMETRIC
Combining these equations with the above, we have
(52)
409
e=p&
f = f = p &
for
ff
= p&
(47)
When these values
are substituted in the first
and the resulting equations are solved
two of equations 7 and 7^ we find
and the
last of equations (47)
reduces to
d
ri2V
a
n2
=
of this equation determines a congruence with the given representation of its developables,* and the middle surface
is
Each solution
given by the quadratures
(54)
and similar expressions
in
y and
2.
When
the values (52) are substituted in (18) the latter becomes
Consequently equation (32) reduces to
a sin ^
2P = ^ =
Referring to equation
(III, 16),
we have:
the focal planes of a congruence is equal to the the lines on the sphere representing the corresponding angle between
The angle between
developables.
This result
168.
is
obtained readily from geometrical considerations.
for the focal surfaces.
Fundamental quantities
We
shall
make use
of these results in deriving the expressions for the funda
,
mental quantities of the focal surfaces Sl and $2 which are defined by
* This result pp. 342344.
is
due to Guichard, Annales de
I
Ecole Normale, Ser.
3,
Vol.
VI
(1889),
410
RECTILINEAR CONGRUENCES
these and (54)
From
we get
The
from
coefficients of the linear elements of
^
and
2,
as derived
these formulas, are
(56)
and
(57)
.
The
tions
directioncosines of the normals to
JT2
,
X^ Yv Z^
r^
2,
2
respectively are
^ and S2 denoted by found from the above equa
and (V, 31)
to
have the values
v
1
.
Si)
ZJ
._
=
3V
/~^
^V
^/^s /./V^
^
_
/}?&gt;
2/^,
i\
\ d(u, v)
^
and similar expressions for Y and Z If these equations be differ of (V, 22), entiated, and the resulting equations be reduced by means they can be put in the form
.
t
{
K
==
"a^"
\\
fi2Vax
^^ "VL^^lJ"^
x I a*t_^/22Vwr \ dv^lif dv
Til2/
FOCAL SURFACES
From
these expressions and (55)
411
we
obtain
^ du
(58)
~
du
V^ \du
^2
o,
D[
=
Y dv du y du cv ^ ^1 Mi = _ ^ ?i Mi =
~2 ^r
=
"^7
*
A"=
and
the foregoing formulas we derive the following expressions for the total curvature of Sl and of Sz
From
:
22V
{1J
(60)
EXAMPLES
1. If upon a surface of reference S of a normal congruence the curves orthog onal to the lines of the congruence are defined by 0(u, u) = const., and 6 denotes the angle between a line of the congruence and the normal to the surface at the
2 = AiF(0) where the differential parameter point of meeting, then sin with respect to the linear element of S. Show that 6 is constant along a line
is
formed
= const.
only
2.
when
the latter
is
a geodesic parallel. a surface, namely
,
,
When
in the point equation of
c2
,
du cv
n  + a c0 + 6 30 = 0,
du
cv
a or
3.
6 is
zero, the coordinates of the surface can be found
by quadratures.
tetrahedral surface (Ex. face Si or 5i is a curve.
4.
Find the derived congruences of the tangents to the parametric curves on a the sur 2, p. 267), and determine under what conditions
Find the equation of the type
given by
(41).
(39)
which admits as solutions the quantities
*i, yi, zi
5. When a congruence consists of the tangents to the lines of curvature in one system on a surface, the focal distances are equal to the radii of geodesic curvature of the lines of curvature in the other system.
412
6.
RECTILINEAR CONGRUENCES
Let S be a surface referred to its lines of curvature, let i and s 2 denote the = const, and u = const, respectively, ri and r2 their radii of curvature, and RI and JR 2 their radii of geodesic curvature for the second
;
arcs of the curves v
first
focal sheet Si of the congruence of tangents to the curves v
=
const, the linear
element
is
reducible to
2
hence the curves
7.
Si
=
const, are geodesies.
is
Show
that
2t
of Ex. 6
r\ so
developable
when
n
=/(si), and determine the
most general form of
8.
that 2i shall be developable.
Determine the condition which p must satisfy in order that the asymptotic on either focal surface of a congruence shall correspond to a conjugate system on the other, and show that in this case
lines
where
9.
denotes the angle between the focal planes.
In order that the focal surfaces degenerate into curves,
it is
necessary and
sufficient that the spherical representation satisfy the conditions
12
{
\
du
10.
={
cv
\
12
\
~ =
(
\
12
1
\
1
)
2
}
Show
that the surfaces orthogonal to a normal congruence of the type of
Ex. 9 are cyclides of Dupin.
11.
A necessary and sufficient condition that the
second sheet of the congruence
developable
is
of tangents to a family of curves on a surface
S be
that the curves
be plane.
169. Isotropic congruences. isotropic congruence is one whose focal surfaces are developables with minimal edges of regression.
An
In
31 we saw that
H=
is
a necessary and sufficient condition
that a surface be of this kind.
Referring to (56) and
(57),
we
see
that
we must have
From (54) it is seen that if p were zero the middle surface would be a point, and from (55) that if the expressions in parentheses were zero the surfaces Sl and $2 would be curves. Consequently
(61)
&lt;
= g=
0.
Conversely,
if this
condition be satisfied,
S and
l
S.2
are isotropic
developables. isotropic congruence opables are represented on the sphere by minimal lines.
Hence an
is
one whose devel
ISOTROPIC CONGRUENCES
In consequence of (61)
413
we
have, from (52),
and since
(62)
f+f also
is
is
zero, it follows that
dxdX+ dydY+ dzdZ= 0.
zero, so that all the lines of striction lie
Therefore r
on the
Since (61) is a consequence of (62), we have the following theorem of Ribaucour,* which is sometimes taken for the definition of isotropic congruences
middle surface.
:
All the lines of striction of an isotropic congruence
dle surface ; and, conversely,
lie
on the mid
when
is
all the lines
;
middle surface, the congruence
face corresponds
linear elements.
to the
isotropic
of striction lie on the moreover, the middle sur
spherical representation with orthogonality of
Ribaucour has established also the following theorem f
:
TJie
middle envelope of an isotropic congruence
is
a minimal surface.
Since the minimal lines on the sphere are parametric, in order to prove this theorem it is only necessary to show that on the
envelope of the middle planes, denotes the corresponding lines form a conjugate system. If the distance of the middle plane from the origin, the condition
middle envelope, that
is,
the
W
necessary and sufficient that the parametric lines be conjugate is that satisfy the equation
W
(63)
r
+
&lt;^0
= 0.
By
definition
and with the aid of (V, 22) we
find
ft
du dv
cu dv
o2
Since equation (53) reduces to
satisfies

+
/&gt;&lt;^=0,
the function
W&gt;
(63).
* Etude des Elasso ides ou Surfaces a Courbure Moyenne Nulle, Memoires Couronnts t L.c., p. 31. par r Academic de Belgique, Vol. XLIV (1881), p. 63.
414
RECTILINEAR CONGRUENCES
Guichard* proposed and solved
170. Congruences of Guichard.
the problem
:
To determine the congruences whose focal surfaces are met by
developables in the lines of curvature.
the
With Bianchi we
call
them congruences of
Gruichard.
remark that a necessary and sufficient condition that a con gruence be of this kind is that Fl and F2 of 168 be zero. From (56) and (57) it is seen that this is equivalent to
We
Comparing
this result
with
78,
we have
the theorem:
of a con gruence meet the focal surfaces in their lines of curvature is that the congruence be represented on the sphere by curves representing also
necessary
the asymptotic lines on a pseudospherical surface.
A
and
sufficient condition that the developables
In this case the parameters can be so chosen thatf
&lt;F=^=1,
c?
=
COSQ),
where
co is
a solution of
= sin
dudv
In this case equation (53)
(65)
is
ft).

= p cos
&lt;0.
In particular, this equation is satisfied by X, F, in (54), we have replace p by
X
Z (V,
22).
If
we
consequently, for the congruence determined by this value of the middle surface is a plane.
/&gt;,
From (55) it follows that the lines of the to the lines of curvature v const, on *Sy
congruence are tangent
=
Consequently they are
(64).
*L.c., p. 346.
f
This
is
the only real solution of
CONGRUENCES OF GUICHARD
call it
415
parallel to the normals to one of the sheets of the evolute of
(cf.
Sl
74)
;
2
.
X
Hence the conjugate system on 2 t
is
corre
represented on the sphere by the same lines as the developables of the congruence. Referring to (VI, 38), we see that condition (64) is equivalent to
sponding to the lines of curvature on S^
where the Christoffel symbols are formed with respect
element of
to the linear
are the conditions that the parametric Surfaces with a conjugate curves (cf. 85). X of geodesies were studied by Voss, * and on this account system are called surfaces of Voss. Since the converse of the above results
true,
2 r But these on 2 be geodesies
is
we have
the following theorem of Guichard
:
and sufficient condition that the tangents to the lines curvature in one family of a surface form a congruence of of Guichard is that one sheet of the evolute of the surface be a sur
face of Voss, and that the tangents constituting the congruence be those which are parallel to the normals to the latter.
If
A necessary
W
to the surface of
(cf.
denotes the distance from the origin to the tangent plane is a solution of equation (65) Voss 2 X then l Hence W^ + Kp is a solution of this equation, provided K 84).
l
,
W
be a constant.
since the tangent plane to 2 X passes through the corresponding point of Sv the above result shows that a plane normal to the lines of the congruence, and which divides in con
But
stant ratio the segment between the focal points, envelopes a sur face of Voss. In particular, we have the corollary
:
The middle envelope of a congruence ofGruichardis a surface of Voss.
171. Pseudospherical congruences.
The
lines joining correspond
ing points on a pseudospherical surface S and on one of its Backhand transforms S1 (cf. 120) constitute an interesting congruence. We
between corresponding points is constant, and that the tangent planes to the two surfaces at these points meet under constant angle. From (32) it follows that the distance
recall that the distance
between the limit points also
is
constant.
(1888), pp. 95102.
*Miinchener JSerichte, Vol. XVIII
416
Conversely,
KECTILINEAB CONGRUENCES
is
gruence
the angle between the focal planes of a con constant, and consequently also the angle 6 between
when
the parametric lines on the sphere representing the developables, we have, from (V, 4),
111112
i
Furthermore, if the distance between the focal points 2 have p = a, and by (60) gjn Q
is
constant, we
K =K =
^
*
"4^"
Hence the two
Congruences
called
focal surfaces
have the same constant curvature.
first studied by Bianchi.* He them pseudospherical congruences. In order that the two focal surfaces of the congruence be Back
of this
kind were
lund transforms of one another,
curvature correspond.
the equation of these lines
is
it is
necessary that their lines of
It is readily
found that for both surfaces reducible by means of (66) to
12V f 12V
{
is\2j
dw
~[^ + f\is +
H2V
2
n
n^v !
2
2
)
J
surface
Moreover, the differential equation of the asymptotic lines on each 2 is dv? 0. Hence we have the theorems: ^/di)
On
the focal surfaces of a pseudospherical congruence the lines of
curvature correspond,
and
likewise the asymptotic lines.
The focal surfaces of a pseudospherical congruence are Backlund
transforms of one another.
EXAMPLES
any whatsoever, and likewise the surface of reference, a condition necessary and sufficient that a congruence be
1.
When
is
the parameters of a congruence are
isotropic
~~ f + f =
e
g
2^
^
necessary and sufficient condition that a congruence be isotropic is that two points on each line at an equal constant distance from the middle surface shall describe applicable surfaces.
2.
A
the locus of
3.
Show
that equation (65) admits
is
and
3u
dv
as solutions.
Prove that
in
each
case one of the focal surfaces
a sphere.
pp. 161172; also Lezioni, Vol.
I,
*Annali, Ser.
2,
Vol.
XV (1887),
pp. 323, 324.
JFCONGRUENCES
4.
is
41T
Determine
all
the congruences of Guichard for which one of the focal surfaces
a sphere.
5.
When
a surface
is
referred to
its lines
of curvature, a necessary and
suffi
cient condition that the tangents to the curves v of Guichard is a/1
=
const, shall
form a congruence
3u\^
Determine the surfaces which are such that the tangents curvature in each system form a congruence of Guichard.
6.
to the lines of
172. TFcongruences. We have just seen that the asymptotic lines on the focal surfaces of a pseudospherical congruence correspond the same is true in the case of the congruences of normals to a
;
124). For this reason this property are called Wcongruences. erties of these congruences.
JFsurface
(cf.
all
congruences possessing
shall derive other
We
prop
The condition
that asymptotic lines correspond,
namely
takes the following form in consequence of (58) and (59):
22V
Hence from (60) it follows that a necessary for a JFcongruence is
and
sufficient condition
In order to obtain an idea of the analytical problem involved in the determination of TFcongruences, we suppose that we have two surfaces S referred to their asymptotic lines, and inquire under
,
what conditions the lines joining corresponding points on the surfaces are tangent to them. We assume that the coordinates of the surfaces are defined* by means of the Lelieuvre formulas (cf. 79), thus:
dx
dx
du
du
(68)
dx_
du
dx
dv
dv
du
du
du
~dv
dv
*Cf. Guichard, Comptes Rendus, Vol.
CX
(1890), pp. 126127.
418
RECTILINEAR CONGRUENCES
y,
z,
and similar equations in
vv
i&gt;
y,
and
z.
The
functions v v
2
,
v s respectively are solutions of equations of the
form
(69)
dudv
and they are such that
(70)
v?
+
l
+
vl
= a,
vl
+
v$
+
vl
= a,
wliere a and a are defined by
J5T
(71)
=,
,,
a
K = ~. a
2
Since v^ v^ v s and v r vz v s are proportional to the directioncosines of the normals to S and S, the condition that the lines
joining corresponding points be tangent to the surfaces
S and
S
is
v^x
z)+
v z (y
y}+ v^(z  z) =
v
z z
0.
Hence
x x
y
where
value,
w
we
denotes a factor of proportionality. In order to find notice that from these equations follow the relations
2
its
(2
/&gt;)
= ^(x  x)*=
2
7?i
2(^ =w
3
a 2)
iy&gt;
where ^ denotes the angle between the focal planes. If this value of 2p and the values of and from (71) be substituted in (67), = 1. We take w=l, thus fixing the signs of it is found that m*
K
K
i&gt;
i/j,
2,
i&gt;
3,
and the above equations become
x
(72)
x
= V&
vfa
y
y
=
v^t
*&,
z
 * = W v v r
2
If the first of these equations
be differentiated with respect to w,
the result
is
reducible by (68) to
JFCONGRUENCES
419
Proceeding in like manner with the others, and also differentiating with respect to v, we are brought to
/
7
/
(73)
.
= 1,2,8)
where
Z
and & are factors of proportionality
to be determined.
If the first of these equations
and
in the reduction
we make
be differentiated with respect to v, use of the second and of (69), we find
In like manner,
if
tiated with respect to u,
the second of the above equations be differen we obtain
Since these equations are true for i=l, parentheses must be zero. , This gives
2,
3,
the quantities in
x
In accordance with the
last
=^
+
cl
3k
dv
du
we put
=
and the others become
a
,
IQOT
*
i
Hence equations
(69)
may
be written
Bdudv
i
dudv
l
dufo\0.if \
from which it follows that l is a solution of the first of equa tions (69) and l/0 l of the second. Moreover, equations (73) may now be written in the form
0,
v&lt;
du
dv
420
RECTILINEAR CONGRUENCES
if Q l be a known solution of the first of equations (69), we obtain by quadratures three functions v which lead by the quadra tures (68) to a surface S. The latter is referred to its asymptotic lines and the joins of corresponding points on S and $ are tangent
Hence
f,
to the latter.
And
so
we have
:
If a surface S
be referred to its asymptotic lines,
and
the equations
of the surface be in the Lelieuvre form, each solution of the corre
sponding equation
ffQ
= \0
S and S
are
dudv
determines a surface S, found by quadratures, such that the focal surfaces of a Wcongruence.
Comparing
(74) with (XI, 13),
we
i
see that
if
we put
^1=^1.
yi=
i=^8
,
the locus of the point (x^ y^ zj corresponds to S with orthogo to the nality of linear elements. Hence v v v 2 v s are proportional of an infinitesimal deformation directioncosines of the generatrices
of
,
so that
we have
:
focal surface of a Wcongruence admits of an infinitesimal deformation whose generatrices are parallel to the normals to the
Each
other focal surface.
Since the steps in the preceding argument are reversible, have the theorem
:
we
trices of
The tangents to a surface which are perpendicular to the genera an infinitesimal deformation of the latter constitute a Wto the
congruence of the most general kind ; and the normals
surface are parallel
to the
other
generatrices of the deformation.
In his study of surfaces corre with orthogonality of linear elements Ribaucour consid sponding ered the congruence formed by the lines through points on one surface parallel to the normals to a surface corresponding with the
173. Congruences of Ribaucour.
calls such a congruence a con and the second surface the director surface. gruence of Ribaucour, In order to ascertain the properties of such a congruence, we
former in this manner.
Bianchi *
recall the results of
153.
Let S be taken for the surface of
l
*Vol.
II, p. 17.
CONGRUENCES OF RIBAUCOUR
reference,
421
If the latter
and draw
its
lines parallel to the
lines, it
normals to
S.
be referred to
asymptotic
follows from (XI, 6) that
civ
du
HK
du dv
9
=y ~^
dx l
dX
Since these values satisfy the conditions
the ruled surfaces
since also
u
= const., v = const,
equal to zero,
p^+ p 2 But the parametric curves on 8 form a conjugate system when the asymptotic lines on S are parametric. Hence we
l
is
S
is
are the developables. And the middle surface of the
t
congruence.
have the theorem
:
The developable surfaces of a congruence of Ribaucour cut the middle surface in a conjugate system.
Guichard
the
first
*
ences of Ribaucour.
proved that this property is characteristic of congru In order to obtain this result, we differentiate
make use
of equations (54) with respect to v, and in the reduction of the fact that and p satisfy equations (V, 22 ) and
X
(53) respectively.
This gives
/isyy*.
dv
is log p \
fi2\as.
\\
}/
du
iu
and similar equations in y and z it follows that a and sufficient condition that the parametric curves form necessary
this
v
From
v
a conjugate system
is
^
f!2V
1
d
T12V
J
du\
S
to\ 2
When
this condition is satisfied
by a system of curves on the
sphere, they represent the asymptotic lines on a unique surface S, whose coordinates are given by the quadratures (VI, 14)
*Annales
L
Ecole Nonnale, Ser.
3,
Vol.
VI
(1889), pp. 344, 345.
422
RECTILINEAR CONGRUENCES
for
and similar expressions
with
(54),
y and
z.
Combining these equations
dx
~
we
find that
_
dx
_ =
du
i
^\ 3x l dx du dv
Z
**
^ dx
l
=
^\ dx v dx =
dv du
^~dv~dv
of linear elements,
Hence S and S correspond with orthogonality
and the normals gruence. Hence
to the
:
former are parallel to the lines of the con
A necessary and sufficient condition that the developables of a congruence cut the middle surface in a conjugate system is that their representation be that also of the asymptotic lines of a surface, in which
case the latter
and
the
middle surface correspond with orthogonality
of linear elements.
EXAMPLES
1.
When
metric curves are asymptotic lines.
is
the coordinates of the unit sphere are in the form (III, 35), the para Find the IFcongruences for which the sphere
one of the focal sheets.
2.
Let
vi
=fi(u)
+
0i (w),
where /; and

&lt;/&gt;
t
and and
i
=
1, 2, 3,
be three solutions of the
first
are functions of u and u respectively, of equations (09), in which case X = 0,
dle surface
be unity. Show that for the corresponding ^congruence the mid a surface of translation with the generatrices u = const., v = const., that the functions / and 0, are proportional to the directioncosines of the binorlet 61 in (74)
is

t
inals to these generatrices,
and that the intersections of the osculating planes of
these generatrices are the lines of the congruence.
3.
Show
that isotropic congruences and congruences of Guichard are congru
ences of Ribaucour.
4.
A
mal
is
necessary and sufficient condition that a congruence of Ribaucour be nor that the spherical representation of its developables be isothermic.
to quadrics
5.
The normals
and
to the cyclidesof
Dupin constitute congruences
is
of Ribaucour.
6.
When
Show
is
the middle surface of a congruence
is
plane, the congruence
of the
Ribaucour type.
7.
helicoid,
8.
that the congruence of Ribaucour, whose director surface is a skew a normal congruence, and that the normal surfaces are molding surfaces.
Show
be normal
is
that a necessary and sufficient condition that a congruence of Ribaucour that the director surface be minimal.
GENERAL EXAMPLES
Through each line of a congruence there pass two ruled surfaces of the con gruence whose lines of striction lie on the middle surface their equation is
1
. ;
edu*
+ (f + f
)dudv
they are called the
mean
ruled surfaces of the congruence.
GENERAL EXAMPLES
2.
423
sphere by an orthogonal system of real
ruled surfaces of a congruence are represented on the lines, and that their central planes ( 105) bisect the angles between the focal planes. Let u = const. v = const, be the mean 167. ruled surfaces and develop a theory analogous to that in
that the
,
Show
mean
3. If the two focal surfaces of a congruence intersect, the intersection is the envelope of the edges of regression of the two families of developable surfaces of
the congruence.
4. If a congruence consists of the lines joining points on two twisted curves, the focal planes for a line of the congruence are determined by the line and the tangent to each curve at the point where the curve is met by the line.
5. In order that the lines which join the centers of geodesic curvature of the curves of an orthogonal system on a surface shall form a normal congruence, it is necessary and sufficient that the corresponding radii of geodesic curvature be func
tions of one another, or that the curves in one family have constant geodesic curvature.
6. Let S be a surface whose lines of curvature in one system are circles; let C denote the vertex of the cone circumscribing S along a circle, and L the corre sponding generator of the envelope of the planes of the circles a necessary and sufficient condition that the lines through the points C and the corresponding lines L
;
form a normal congruence is that the distance from C to the points of the correspond ing circle shall be the same for every circle if this distance be denoted by a, the
;
radius of the sphere
is
given by
_ p /2
/
jn
\
a 2\
to the arc of the curve of
where the accent indicates differentiation with respect
centers of the spheres.
7.
Let
S
be a surface referred to
its lines
of curvature, Ci
and
C2
the centers
of principal normal curvature at a point, GI and G 2 the centers of geodesic curva ture of the lines of curvature at this point; a necessary and sufficient condition
of Pg u
that the line joining C2 and G\ form a normal congruence or that one of these radii be a constant.
,
is
that p 2 be a function
8. Let S be a surface of the kind defined in Ex. 6; the cone formed by the normals to the surface at points of a circle A is tangent to the second sheet of the
evolute of
the vertex
S
in a conic
T
(cf.
132).
Show
that the lines through points of T and
C
of the cone
which circumscribes 8 along
the plane of F.
A
generate a normal con
gruence, and that
9.
C
lies in
Given an isothermal orthogonal system on the sphere for which the linear
is
Z
element
_
*
+
2 cto )
;
on each tangent to a curve v = const, lay off the segment of length X measured from the point of contact, and through the extremity of the segment draw a line parallel to
the radius of the sphere at the point of contact
.
Show that this congruence is iso tropic.
its
a congruence is iso tropic and 10. 35), equation (53) reduces to (III,
When
directioncosines are of the
form
8uBv
(1fww)
Show
that the general integral
p
is
= 2 O0(v) 
vf(u)](l
+
uv)
v respectively.
where / and
are arbitrary functions of u of the middle surface.
and
Find the equations
424
11.
RECTILINEAR CONGRUENCES
Show
that the intersections of the planes
(1

M 2 )x
v 2 )z
f i (1
+ w2
)
y
(1

i(l
+
v z )y
+ +
2 uz
+
4/(w)
=
0,
2vz
+40(u)=
;
constitute an isotropic congruence, for which these are the focal planes that the locus of the midpoints of the lines joining points on the edges of regression of the developables enveloped by these planes is the minimal surface which is the middle
envelope of the congruence, by rinding the coordinates of the point in which the tangent plane to this surface meets the intersection of the above planes.
12. Show that the middle surface of an isotropic congruence is the most general surface which corresponds to a sphere with orthogonality of linear elements, and that the corresponding associate surface in the infinitesimal deformation of the
sphere
13.
is
the minimal surface adjoint to the middle envelope.
to the
Find the surface associate
middle surface of an isotropic congruence
when
is
the surface corresponding to the latter with orthogonality of linear elements a sphere, and show that it is the polar reciprocal, "with respect to the imaginary
2 2 2 sphere x f y f z of the congruence.
+1=
0, of
the minimal surface adjoint to the middle envelope
14. The lines of intersection of the osculating planes of the generatrices of a surface of translation constitute a ITcongruence of which the given surface is the middle surface if the generatrices be curves of constant torsion, equal but of
;
opposite sign, the congruence
is
normal
to a TFsurface of the type (VIII, 72).
15. If the points of a surface
S be
projected orthogonally upon any plane A,
and
if,
after the latter has been rotated about
any
line
normal
to
it
through a
parallel to the corresponding nor right angle, lines be drawn through points of mals to S, these lines form a congruence of Ribaucour.
16. A necessary and sufficient condition that the tangents to the curves v const. on a surface, whose point equation is (VI, 26), shall form a congruence of Ribaucour is
A
aa_S6
du
dv
dudv
17. Show that the tangents to each system of parametric* curves on a surface form congruences of Ribaucour when the point equation is
where Ui and V\ are functions of u and v respectively, and the accents indicate
differentiation.
18. Show that if the parametric curves on a surface S form a conjugate system, and the tangents to the curves of each family form a congruence of Ribaucour, the same is true of the surfaces Si and S_i, which together with S constitute the focal surfaces of the two congruences.
19.
Show
that the parameter of distribution
is
p
of the ruled surface of a con
gruence, determined by a value of dv/du,
given by
f
P=
1
e
du
+ /du,
f du
f
g dv
GENERAL EXAMPLES
20.
425
Show
by
that the
mean
ruled surfaces
(cf.
Ex.
1)
of a congruence are char
the property that for these surfaces the parameter of distribution has the maximum and minimum values.
acterized
21. If S and SQ are two associate surfaces, and through each point of one a line be drawn parallel to the corresponding radius vector of the other, the developables of the congruence thus formed correspond to the common conjugate system of S and SQ,
22. In order that two surfaces S and SQ corresponding with parallelism of tangent planes be associate surfaces, it is necessary and sufficient that for the and MQ of these sur congruence formed by the joins of corresponding points faces the developables cut S and SQ in their common conjugate system, and that
M
the focal points
M and
MQ form
a harmonic range.
23. In order that a surface S be iso thermic, it is necessary and sufficient that there exist a congruence of Ribaucour of which S is the middle surface, such that the developables cut S in its lines of curvature.
CHAPTER
XIII
CYCLIC SYSTEMS
174. General equations of cyclic systems. The term congruence not restricted to twoparameter systems of straight lines, but is * applied to twoparameter systems of any kind of curves. Darboux
is
has
made a study of these general congruences and Ribaucourf has considered congruences of plane curves. Of particular interest is the case where these curves are circles. Ribaucour has given the
name
cyclic systems to
parameter family of orthogonal surfaces. to a study of cyclic systems.
congruences of circles which admit of a oneThis chapter is devoted
begin with the general case where the planes of the circles associate with the latter envelop a nondevelopable surface S.
We
We
a
moving
trihedral
(
68),
and
for the present
assume that the
parametric curves on the surface are any whatever. As the circles lie in the tangent planes to S, the coordinates
of a point on one of them with respect to the corresponding trihedral are of the form
(1)
a
a, b
+ Rcos0,
b+Rsm0,
0,
where
are the coordinates of the center,
latter to a
R
the radius, and
the angle which the
given point makes with the moving
In
69 we found the following expressions for the projections
of a displacement of a point with respect to the
moving axes
t
:
(")
(dx+%du + ^dv + (qdu + q^v) z du 4 rj^dv + (rdu + r dv] x \ dy +
77
v
+ r dv) y, (pdu + p^dv) z,
(rdu
\dz
* Vol.
+(p du +p 1 dv) y
(qdu+
q v dv) x,
Math.
II, pp. 110; also Eisenhart, Congruences of Curves, Transactions of the Amer. Soc., Vol. IV (1903), pp. 470488. t Memoire sur la theorie generale des surfaces courbes, Journal des Mathtmatiques,
Ser. 4, Vol.
VII
(1891),
117
et. seq.
426
GENERAL EQUATIONS
where the translations f f 1?
,
427
p,q,r&gt;
T;,
^ and
the rotations
p^
qv
,
r^
satisfy the conditions
dp_di_
(3)
__
d__Mi
=
dr
dr.
^Wl
(1)
PH
PhM^flin
(2)
When
the
values
are
substituted
the latter
are
reducible to
J du + J^v cos QdR (dd + rdu+ r^dv) R sin ^, # ^w + ^jrfu + sin 6dR + (dd+rdu + r^dv] R cos 6, (y du + q^dv) (a+R cos (p du f ^^v) (b + R sin 0)

where we have put, for the sake
of brevity,
^
du
(5)
The
conditions that
du \dv]
are reducible, by
(\ = dv (^
means
\du/
du \dv
\du
of (3), to
(6)
_
dv
du
The
point
(7)
directioncosines of the tangent to the given circle at the
(1) are
sin0,
cos0,
0.
Hence the condition that the
be orthogonal to the circle
multiplied respectively by
is
locus of the point, as u and v vary, that the sum of the expressions (4)
the quantities (7) be zero.
This gives
428
CYCLIC SYSTEMS
In order that the system of circles be normal to a family of sur
faces this equation must admit of a solution involving a parameter. Since it is of the form
(9)
the condition that such an integral exist
is
that the equation
be
satisfied
identically.
*
For equation
(8)
this
condition
is
reducible to
In order that this equation be satisfied identically, the expressions in the brackets must be zero. If they are not zero, it is possible that the two solutions of this equation will satisfy (8), and thus
determine two surfaces orthogonal to the congruence of Hence we have the theorem of Ribaucour:
circles.
If
the circles of a congruence are
normal
to
more than two surfaces,
they form a cyclic system.
The equations
consequently
of
condition
that
the
system be cyclic are
dR
.
dR
.
The
total curvature of
S
is
given by
(cf.
70)
*
Equations,
Murray, Differential Equations, p. 257. London, 1888.
p. 137.
New York,
1897; also Forsyth, Differential
THEOREMS OF RIBAUCOUR
429
From this and (5) it is seen that equations (12) involve only functions relating to the linear element of S and to the circle. Hence we have the theorem of Ribaucour:
If the envelope of the planes of the
circles of
a cyclic system
be
deformed in any manner without disturbing
continues
to
the size or position of
the circles relative to the point of contact, the congruence of circles
form a
if
cyclic system.
Furthermore,
we put
t
= tan Q
z*
&gt;
equation
(8)
assumes the Riccati form,
dt
+ (af +a t +
2
a 3 ) du
+ (b/ +
bjt
+
b3 )
dv
= 0,
:
where the # s and 5 s are functions of u and v. Recalling a funda mental property of such equations ( 14), we have
Any four
orthogonal surfaces of a cyclic system meet the circles in
is constant.
four points whose crossratio
Since by hypothesis be replaced by
S
is
nondevelopable, equations (12)
may
du
(13)
^
dv
AB, 
 trf)
JBT
=
0.
By
(5)
the
first
two
of these equations are reducible to
du
(14)
a
The condition
(15)
of integrability of these equations
is
^{ + g,^f g, r(f^J )r (,^. cu du cv dv
1 l
1 1
Instead of considering this equation, by the equation
2 2
we
2
,
introduce a function
&lt;j&gt;
(16)
24&gt;=,K
a &
430
CYCLIC SYSTEMS
must
&lt;/&gt;
and determine the condition which
a and
b the
satisfy.
We
is
take for
expressions obtained by solving (14); that
(17)
Now
the equation (15) vanishes identically, and the only other condition to be satisfied is the last of (13); this, by the substi
tution of these values of
a,
6,
R, becomes a partial differential
equation in
(18)
&lt;
of the
form
du
r ^_r
du dv
7,
_(__r_] \J^
\dudv /
+L
ducv
&lt;&gt;,
+ M ~ + N= 0,
du
where
X,
JHf,
JV denote functions of
f
,
rt,
and
their deriva
tives of the first order.
Conversely, each solution of this equation
lie
gives a cyclic system whose circles
in the tangent planes to S.
EXAMPLES
1.
Let S be a surface of revolution defined by
is
(III, 99),
and
let
Tbe the
trihedral
whose xaxis
function
\f/
(u)
tangent to the curve v = const. Determine the condition which the must satisfy in order that the quantities a, b in (1) may have the values
a
=
_*w_.
6
=
l,
also the expression for R. necessary and sufficient condition that all the circles of a cyclic system whose planes envelop a nondevelopable surface shall have the same radius, is that
2.
and determine
A
the planes of the circles touch their envelope that S be pseudospherical.
3.
S
at the centers of the circles,
and
Let S be a surface referred to an orthogonal system of lines, and let T be With reference to the is tangent to the curve v = const. trihedral the equations of a curve in the tangent plane are of the form x = p cos 0, z = 0, y = p sin 0,
the trihedral whose zaxis
where
in general p
is
a function of
0,
w,
and
is
v.
Show
that the condition that there
0,
be a surface orthogonal to these curves
that there exist a relation between
u,
and v which
satisfies the
equation
U sin
When
this condition is satisfied
stant, there is
f prji
cos
30
by a function which involves an arbitrary con an infinity of normal surfaces. In this case the curves are said to by the planes
in
form a normal congruence.
4.
When
the surface enveloped
is
of the curves of a
normal con
gruence of plane curves
deformed
such a
way
that the curves remain invari
ably fixed to the surface, the congruence continues to be normal.
CYCLIC CONGRUENCES
175. Cyclic congruences.
431
tem constitute a
rectilinear
of the circles of a cyclic sys * congruence which Bianchi has called
The axes
a cyclic congruence. In order to derive the properties of this con gruence and further results concerning cyclic systems, we assume that the parametric curves on S correspond to the developables of the congruence. The coordinates of the focal points of a line of the congruence with reference to the corresponding trihedral are of the form
a,
ft,
p^
a,
ft,
/&gt;
2
.
On
we
coordinates
of
the
focal
the hypothesis that the former are the point for the developable v = const,
(2),
through the
cu
line,
have, from
rft
+
? +?/i
=
0,
cu
 + T)pp +ra =
l
Q.
of similar equations.
Proceeding in like manner with the other point, we obtain a pair All of these equations may be written in the
abbreviated form
(19)
A+
qPl
=0,
Ji PPl =Q,
^+ ?lft =0,
1
J?,^p2 =0,
in consequence of (5). last of equations (13),
(20)
When
it is
these values are substituted in the
found that
Sf=P
lines joining a point
If
pf
circle to the focal points are
Hence the
on the
perpendicular.
we put
thus indicating by 2 p the distance between the focal points, and by 8 the distance between the center of the circle and the midpoint
of the line of the congruence,
we
find that
We
(21)
replace this equation by the two
8
p cos
a.
&lt;r,
R
p sin
cr,
thus defining a function
/5 1
Now we
have
1),
=/E)(coso+l),
/? 2 =/o(cos&lt;r
so that equations (19)
may
be written
A=
(22)
^*
i
o. o
= _ q ip (cos
1),
B = prf (cos ol
1).
*Vol.
II, p. 161.
432
CYCLIC SYSTEMS
of (5) equation (15) can be put in the
By means
form
When
the values (22) are substituted in this equation,
it
becomes
Since by (3) the expression in the first parenthesis is true of the second, and so we have
is
zero, the
same
these are the conditions (V, 67) that the parametric curves on S form a conjugate system. Hence we have the theorem of
But
Ribaucour
:
On the envelope of the planes of the circles of a cyclic system the curves corresponding to the developables of the associated cyclic con
gruence form a conjugate system.
176. Spherical representation of cyclic congruences.
When
the
expressions (22) are substituted in
(6),
we
obtain
do
da.
dp
dp,
Since pq l p l q =t= unless Sis developable, the preceding equations be replaced by may
12
[ /3
(cosol)]=2
a hi)]
/0
{
2
}VH^),
1J
\
(
d
_
cv
[/3(cos
=
2psL
/12V +
a(
l\P\)
i
where the Christoff el symbols are formed with respect to
(24)
(pdu+p
2
l
dv) \(qdu{q l dvY,
S.
the linear element of the spherical representation of
SPHERICAL REPRESENTATION
433
of equations
When
(13),
in like
manner we
J?
2
substitute in the
2
first
two
taking
=p
2
sin
cr
=p
2
(1
cos
2
o),
we
obtain
.
cos
dp
a)
du
p cos o d cos a 1 4 cos a du P cos
1
*
,
=pb
p
o
r
qa,
(1
n+
,
\ p cos o) J7
cos
o
a
cos
From
these equations and (23)
we
find
The
condition of integrability of equations (25)
is
reducible to
obtained from this equation be substi tuted in (25), we find two conditions upon the curves on the sphere in order that they may represent the developables of a cyclic con
If the expression for cos
a
gruence.
isfied,
A
when
particular case is that in which (27) the two conditions are
is
identically sat
121 ri
O"\
(28)
ll
1/121 _
O"^
"
^
n
J
0v
I2J
2
n o^ f 121
ll
if
/
f
f1
121
o"\
/
J
12 /
It is
now our purpose
to
show
that
any system of curves on
the sphere satisfies either set of conditions, all the congruences whose developables are thus represented on the sphere are cyclic.
We
assume that the sphere
a solution p of
is
referred to such a system and that
we have
434
CYCLIC SYSTEMS
the
By
method
of
167, or that hereinafter explained,
we
find the
middle surface of the congruence. Then we take the point on each line at the distance p cos a from the midpoint as the center of the
circle of radius p sin
a and
for
which the
line is the axis.
These
cir
cles
form a
cyclic system, as
we
shall show.
In the
first
place
we determine
the middle surface with reference
2axis coincides
to a trihedral of fixed vertex,
whose
If
with the radius
of the sphere parallel to the line of the
and #axes are any whatever.
#
,
#
,
z
congruence and whose xdenote the coordinates
of the midpoint of a line with reference to the corresponding tri hedral, the coordinates of the focal points are
From
ables
(2) it is
seen that
if
v = const,
and
u = const,
these points correspond to the developrespectively,
we must have
Since pq l p l q =t= 0, the conditions of integrability of these equa tions can be put in the form
(30)
It is readily
found that the condition of integrability of these equa
tions
reducible to (29). It will be to our advantage to have also the coordinates of the
is
point of contact of the plane of the circle with its envelope S. If x, y, Z Q p cos a denote these coordinates with reference to the
above trihedral,
it
follows from (2) that
(z
 p cos v) + pyqx = 0,
o
(z
 p cos a) + p^y  q,x = 0.
CYCLIC CONGRUENCES
435
If these equations be subtracted from the respective ones of (30), the results are reducible, by means of (25), to
(cos
a 1)

+ 2 p cos 1
a
gj
+ p(y  y)  q(x  x) = 0,
Q
(cos
cr
+ 1)
+ 2 p cos
a
V^ y)
ftfo
x)
=
0,
For, the quantities x x, y Q y are the coordinates of the center of the circle with reference to the tri
(26).
which are the same as
hedral parallel to the preceding one and with the corresponding point on S for vertex.
If,
then,
we have
a solution
b
cr
of (25)
and p
of (29), the corre
(22), since the
sponding values of a and
given by (26) satisfy
latter are the conditions that the parametric curves
on the sphere the developables of the congruence. However, we have represent seen that when the values (22) are substituted in (12), we obtain
equations reducible to (25) and (26). Hence the circles constructed as indicated above form a cyclic system.
Since equations (25) admit only one solution (27) unless the con
dition (28)
is satisfied,
we have
the theorem:
With each
tem unless
case there is
cyclic congruence there is associated a
it is
at the
unique cyclic sys same time a congruence of Ribaucour, in which
an
infinity of associated cyclic systems.
Recalling the results of
141,
we have
the theorem of Bianchi *
:
When
the total curvature of a surface referred to its
asymptotic
lines is of the
form

~
[*
it is
the surface generatrix of a congruence of
Ribaucour which
is
cyclic in
an
infinity of ways,
and
these are the only cyclic
congru
ences with an infinity of associated cyclic systems.
In this case the general solution of equations (25)
(31)
is
cos
is
&lt;,=&gt;*
&lt;#&gt;
+ *, +^
where a
an arbitrary constant.
* Vol.
II, p. 165.
436
CYCLIC SYSTEMS
177. Surfaces orthogonal to a cyclic system. In this section we consider the surfaces Sl orthogonal to the circles of a cyclic sys tem. Since the directioncosines of the normals to the surfaces
with reference to the moving trihedral in
174 are
sin 6, cos 0, 0,
the spherical representation of these surfaces is given by the point whose coordinates are these with respect to a trihedral of fixed vertex parallel to the above trihedral. From (2) we find that the
expressions for the projections of a displacement of this point are
+ rdu + ^ dv), sin 6 (dd + r du + r dv), (p du + p^dv) cos 6 f (q du + q dv) sin 0.
cos 0(d0
1
v
Moreover, by means of
(32)
(8), (21), (22),
we
obtain the identity
(
sin
a
(dd
+ r du + r^dv) =
(1 4 cos a)
p
cos 6 f q sin 6)
du
+ (1
Hence the
(33)
cos a) ( p l cos 6
+
&lt;?!
sin 0) dv.
linear element of the spherical representation of
^
is
da*=
T
L

COS
(p cosO
+ q sin 0fdu*
O~
1\
+ COSOcurvature,
if
r
1
Since the parametric curves on the sphere form an orthogonal system, the parametric curves on the surface are the lines of
that this condition
they form an orthogonal system. In order to show is satisfied, we first reduce the expressions (4)
v
for the projections of a displacement of a point of (21), (22), (25), (26), and (32), to
on Sv by means
cos v sin
Cdu
&lt;r
(
,1
cos
a
1
(34)
,
Cdu
cos
&lt;r
sin
sin
&lt;T
i
Ddv \ + cos 07 Ddv \ 1 + cos
cr,
)
Cdu + Ddv,
where we have put
= pi(b + R sin 0)q (a+E cos 0).
1
NORMAL CYCLIC CONGRUENCES
Hence the
(36)
437
linear element of
S
1
is
ds* = 2
it is
du
.
1
COS
+
(T
2
1
D
V
,
+ COS
cr
from which
seen that the parametric curves on Sl form an orthogonal system, and consequently are the lines of curvature. Furthermore, it is seen from (34) that the tangents to the curves
v
= const.,
u
= const, make
i/l
1
tan"
:
with the plane of the
1
tan"
circle the respec
tive angles
..
(37)
COScrX
),
./
!
lfCOSCT\
sin
\
sin
cr
/
\
a
/
But
it
cumference of a
follows from (21) that the lines joining a point on the cir circle to the focal points of its axis make the
angles (37) with the radius to the point.
Hence we have
:
to a cyclic system of the congruence of axes of the circles, correspond developables and the tangents to the two lines of curvature through a point of the surface meet the corresponding axis in its focal points.
The
lines of curvature
to the
on a surface orthogonal
178.
Normal
cyclic congruences.
Since the developables of a
cyclic congruence correspond to a conjugate system on the enve lope S of the planes of the circles, this system consists of the
lines of curvature
this case (cf.
when
83). If, of the trihedral tangent to the lines of curvature, edges
the congruence is normal, and only in under these conditions, we take two of the
we have
and equations (25) become
d
,
03,
"
d
,
.
cr
d
By
a suitable choice of parameters
tt
we have
so that
(39)
if
we put
&&gt;
=
cr/2,
the linear element of the sphere
2 2
.
is
d(r*=
sinW% + cosWv
(
Comparing
of
its lines
this result with
119),
we have
the theorem:
The normals
to a surface 2 with the same spherical representation curvature as a pseudospherical surface constitute the of
only kind of normal cyclic congruences.
438
Since the surface
CYCLIC SYSTEMS
of the planes of the circles have the same representation of their lines of curvature, the tangents to the latter at corresponding points on the two surfaces are parallel. Hence with reference to a trihedral for 2
parallel to the trihedral for
2 and
the envelope
the coordinates of a point on the circle are R cos 0, R sin 6, p, where ft remains to be determined and 6 is given by (32), which can be put in the form
,*(\^
S
cd
H
d(D
(40)
du
n = cos co sm 0,
.
dO
1
da)
=
.
sin
&&gt;
cos
6.
a
dv
dv
du
If
we
express,
by means
of (2), the condition that all displace
ments of
this point
be orthogonal to the line whose direction
cosines are
sin 0, cos 0, 0, the resulting equation is reducible,
by means
of (40), to
sin 9 (R cos
o&gt;
fi
sin
&&gt;
o&gt;
f)
//,
du
&&gt;
cos
(R
sin
f
cos
77^
dv
0.
Hence the
(41)
quantities in parentheses are zero, from which
i;
we
o&gt;.
obtain
R=
cos
&)
h
T] I
sin w,
^
=
f sin
o&gt;
+
rj l
cos
When, in particular, 2 is 2 I/a we have (VIII, 22)
,
a pseudospherical surface of curvature
f
so that
R= a and /x and the envelope of
(cf.
= a cos = 0. Hence
&),
rj l
= a sin
&&gt;,
the circles are of constant radius
is
their planes
the locus of their centers
is sat 174). in this follows from (13) that is constant. Moreover, case p^ and p 2 as defined in 175, are the principal radii of the surface, which by (20) is pseudospherical. When these values
Ex.
2,
Conversely, when
the latter condition
&lt;
isfied, it
R
,
are substituted in (36) and (33), ment of each orthogonal surface
it
is
is
found that the linear
ele
ds*
= a? (cos
2
6 du* f sin 2 6 dv z ),
and
(42)
of its spherical representation
d&lt;r*=sm*0du*+
cos
2
&lt;W.
Hence
these orthogonal surfaces are the transforms of
(
2 by means
of the Bianchi transformation
119).
PLANES OF CIRCLES TANGENT TO A CUKVE
The expression
(42)
is
439
the linear element of the spherical rep resentation of the surfaces orthogonal to the circles associated with
be pseudospherical or not, whose spherical representation given by (39). Since these orthogonal surfaces have this representation of their lines of curvature, they are of the
any surface 2, whether
is
it
same kind
as 2.
We
have thus for
all
surfaces with the same rep
resentation of their lines of curvature as pseudospherical surfaces, a transformation into similar surfaces of which the Bianchi trans
formation
is a particular case transformation.*
;
we
call it
a generalized Bianchi
179. Cyclic systems for
circles is
which the envelope
of the planes of the
a curve.
We consider now the particular cases which
have
for
been excluded from the preceding discussion, and begin with that which the envelope S of the planes of the circles is a curve C.
take the moving trihedral such that its zyplane, as before, that of the circle, and take the zaxis tangent to C. If s denotes the arc of the latter, we have
is
We
ds
=
f;
du
+
and by
(43)
^dv,
??
=
rj 1
=
0,
(3)
r
^r^ = Q,
(16)
it
rfxftf
= 0.
are functions of
&lt;/&gt;
From
(14), (15),
and
follows that a and
s,
so that these equations
(44)
may
.K
2
be replaced by
2
2
=a +&
on the sphere represent the developables of the congruence, the conditions (19) must hold. But from (5), _Q (15), and (43) we obtain
If the parametric curves
^.
_^
If the values
from
(19) be substituted in this equation,
we
have,
from
(43),
^^=0.
focal surfaces coincide.
If
Hence the
we put
P
in (19)
= Pi=P*
we
obtain
and substitute
in the last of (12),
2
1
(^+^ )(^ ^)==0.
*Cf.
American Journal, Vol.
XXVI
(1905), pp. 127132.
440
CYCLIC SYSTEMS
of
The vanishing
p
pq
l
p
l
q
is
the condition that there be a single
infinity of planes,
which case we exclude for the present. Hence
the developables of the cyclic congruence are
=
iR
;
that
is,
imaginary. Instead of retaining as parametric curves those representing the take the arc of C developables, we make the following choice.
We
for the parameter u consequently f =1, also, we have, from (3),
;
^=0.
Since
77
=^ =
x
.hence we
dq
2MB*
parameter v so that
may
it
choose the
p=
0,
^=1.
From
(3)
follows, furthermore, that
dr
i=
of
T
&gt;
BV
= *
which the general integral
is
q=U
where U^ and
l
cos v
+U
2
sin
v,
r
=
U
l
sin v f
U
z
cos
v,
U
2
are arbitrary functions of u.
From
(5)
we have
A=
j&gt;"(u)+\rl,
^=0,
is
A = 7T #y
U
so that the third of equations (12)
(*"+!)
reducible by (44) to
cv
d
U
sin v
cos v
Hence
if
we
6,
(45) gives
take for a any function of u denoted by and R follows directly from (44).
r
&lt;f&gt;
(u),
equation
180. Cyclic systems for
which the planes
of
the circles pass
through a point.
If the planes of the circles of a cyclic system a point 0, we take it for the origin and for the pass through vertex of a moving trihedral whose zaxis is parallel to the axis
of the circle
under consideration.
In this case equations (14)
may
(46)
be replaced by
2
A&gt;
=
2
tf
+6
2
*,
in
where
c
denotes a constant.
But
this is the condition that all the
it
circles are
cut orthogonal to a sphere with center at 0, or
PLANES OF CIKCLES THROUGH A POINT
diametrically opposite points, or pass through 0, according as positive, negative, or zero. Hence we have the theorem
:
441
c is
If the planes of the
circles of
a cyclic system pass through a point,
the circles are orthogonal to a sphere with its center at the point, or meet the sphere in opposite points, or pass through the center.
From
geometrical considerations
is
we
see that the converse of this
theorem
true.
When
by
(21)
(47)
c in (46) is zero all
the circles pass through 0.
Then
we have
a
=
p sin
&lt;r
cos 6,
b
=
p sin a sin
6,
and equations (26) become
(cos
or
1)
^
=
.
2 cos
a
\
f
,
+ sin
a
(p sin 6
q cos
(cos
&lt;r
+1)

SL.
2cos&lt;rj
f
\smo(p l sm0
These equations are obtained likewise when we substitute the values (47) in equations (22) and reduce by means of (25) and (32).
Because of (22) the function p given by (26)
arid therefore p
is
given by (48)
is
a solution.
But
a solution of (29), the solution 6 of
of Bianchi *
:
(32) involves a parameter.
Hence we have the theorem
Among all the cyclic congruences with the same spherical repre sentation of their developables there are an infinity for which the circles of the associated cyclic system pass through a point.
If
we
take the line through
the zaxis of the trihedral, equation (11)
tion
= TT,
and the center of the circle for must admit of the solu and consequently must be of the form
In order that
zero and the system cyclic. combine this result with the preceding theorem to obtain the following:
L and
M must be
this equation
admit of a solution other than
TT,
both
We
A
to
twoparameter family of circles through a point and orthogonal any surface constitute a cyclic system, and the most general spher
ical representation of the developables of a cyclic congruence is afforded
by the representation of the axes of such a system of
* Vol.
II, p. 169.
circles.^
t Bianchi, Vol. II, p. 170.
442
CYCLIC SYSTEMS
consider finally the case where the planes of the circles depend upon a single parameter. If we take for moving axes the
tangent, principal normal, and binormal of the edge of regression of these planes and its arc for the parameter w, we have
We
and comparing (V, 50) with
p
(2),
we
see that
&lt;7
T
= 0,
r
=p
of the
where p and r are the
edge of regression.
b da A = hi ,
radii of first
and second curvature
Now
du
p
A= 8a dv
,
;
= B _#,hdu
p
r&gt;
2?!
= *b
dv
The equations (12) reduce to two. One of the functions a, b may be chosen arbitrarily then the other and R can be obtained by the solution of partial differential equations of the first order.
EXAMPLES
1.
Show
is
helicoid
cyclic,
that a congruence of Ribaucour whose surface generator and determine the cyclic systems.
is
the right
2. congruence of Guichard is a cyclic congruence, and the envelope of the planes of the circles of each associated cyclic system is a surface of Voss.
A
3. The surface generator of a cyclic congruence of Ribaucour surface of the planes of the circles of each associated cyclic system.
4.
is
an associate
If
S
is
tion as a pseudospherical surface,
a surface whose lines of curvature have the same spherical representa and Si is a transform of S resulting from a gen
eralized Bianchi transformation
( 178), the tangents to the lines of curvature of 81 pass through the centers of principal curvature of S.
5. When the focal segment of each line of a cyclic congruence is divided in constant ratio by the center of the circle, the envelope of the planes of the circles is a surface of Voss.
6.
The
circles of the cyclic
system whose axes are normal to the surface S,
defined in Ex. 11, p. 370, pass through a point, and the surfaces orthogonal to the circles are surfaces of Bianchi of the parabolic type.
7. If the spheres with the focal segments of the lines of a congruence for diameters pass through a point, the congruence is cyclic, and the circles pass
through the point.
8.
Show
that the converse of Ex. 7
is
true.
GENERAL EXAMPLES
GENERAL EXAMPLES
1.
443
Determine the normal congruences of Ribaucour which are
If the
cyclic.
2.
Voss whose
envelope of the planes of the circles of a cyclic system is a surface of conjugate geodesic system corresponds to the developables of the asso
ciated cyclic congruence, any family of planes cutting the focal segments in con stant ratio and perpendicular to them envelop a surface of Voss.
3. necessary and sufficient condition that a congruence be cyclic is that the developables have the same spherical representation as the conjugate lines of a sur face which remain conjugate in a deformation of the surface. If the developables
A
of the congruence are real, the
4.
deforms of the surface are imaginary.
The planes
Ribaucour touch their respective envelopes
straight line.
of the cyclic systems associated with a cyclic congruence of in such a way that the points of con
tact of all the planes corresponding to the
same
line of the
congruence
lie
on a
5. If the spheres described on the focal segments of a congruence as diameters cut a fixed sphere orthogonally or in great circles, the congruence is cyclic and the circles cut the fixed sphere orthogonally or in diametrically opposite points.
6. If one draws the circles which are normal to a surface S and which cut a fixed sphere S Q in diametrically opposite points or orthogonally, the spheres described on the focal segments of the congruence of axes as diameters cut SQ in
great circles or orthogonally.
7.
Determine the cyclic systems of equal
circles
whose planes envelop a devel
opable surface.
8. Let Si be the surface defined in Ex. 14, p. 371, and let S be the sphere with center at the origin and radius r. Draw the circles which are normal to Si and which cut S orthogonally or in diametrically opposite points. Show that the of the axes of these circles is a normal congruence, and that the
cyclic congruence coordinates of the
normal surfaces are of the form
(
a?e
a

2
(r?
[1
L2
+
2
)
K) e
cos 6
f
77
sin 8
T
\Xi
j
+ fJL ja2 e~ (
(T;
+
K )(P
I
sind

rj
cose]
X
2
+
tX,
or in
)
r2 or + r2 according as the circles cut K is equal to diametrically opposite points, and where t is given by
where
,
S orthogonally
(
2e
2a( [1
\a
a
a
2
(i?
I
a K) e
^)
[
cos 6
)
+
"1
r\
sin 6
sin
du
_
[1
9.
(
a?e
~\
2
j
(j
4 K)
e"
sin 6
j
t\
cos
cos u dv.
Show
same spherical representation of their surface S referred to in Ex. 14, p. 371.
that the surfaces of Ex. 8 are surfaces of Bianchi which have the lines of curvature as the pseudospherical
Ex. 8 are surfaces 10. Show that the surfaces orthogonal to the cyclic system of Bianchi of the parabolic type. of
444
11.
CYCLIC SYSTEMS
Let S be a surface referred to an orthogonal system, and
is
let
T be
the trihe
.
dral whose xaxis
tangent to the curve u
=
=
const.
0,
The equations
z
x
=
p(l
+
cos0),
y
=
/&gt;sin0
define a circle normal to S.
Show
that the necessary and sufficient conditions that
the circles so defined
form a
cyclic system are
cu
when an orthogonal
necessary and sufficient condition that a cyclic system remain cyclic surface S is deformed is that S be applicable to a surface of revolution and that
12.
A
where
13.
c is
a constant and the linear element of S
is
ds2
=
du 2
+
2
(w) dv* (cf
.
Ex.
11).
Determine under what conditions the lines of intersection of the planes of the circles of a cyclic system and the tangent planes to an orthogonal surface form a normal congruence.
M
Let Si and S 2 be two surfaces orthogonal to a cyclic system, and let MI and be the points of intersection of one of the circles with Si and S 2 Show that the normals to Si and S 2 at the points MI and 2 meet in a point equidistant
14.
2
.
M
M
points, and show that Si and S 2 constitute the sheets of the envelope of a twoparameter family of spheres such that the lines of curvature on Si and S 2
from these
correspond.
15. Let
S
variable radius
spheres.
be the surface of centers of a twoparameter family of spheres of JR, and let Si and S 2 denote the two sheets of the envelope of these
Show that the points of contact MI and 2 of a sphere with these sheets are symmetric with respect to the tangent plane to S at the corresponding point M. Let S be referred to a moving trihedral whose plane y = is the plane 2 and
M
MiMM
,
let
if
the parametric curves be tangent to the x and yaxes respectively. Show that ff denotes the angle which the radius MMi makes with the xaxis of the trihedral,
the lines of curvature on Si are given by
sin
&lt;r
(sin
&lt;rp
r cos
&lt;r)
du 2
+
in[qi
\
I sin
&lt;r
)
dv 2
dv/
H fllfl
)
(cos
&lt;rri
+ p sin
&lt;r)
\dudv
=
0.
16. Find the condition that the lines of curvature on S! and S 2 of Ex. 15 corre spond, and show that in this case these curves correspond to a conjugate system on S. 17. Show that the circles orthogonal to two surfaces form a cyclic system, pro vided that the lines of curvature on the two surfaces correspond. 18.
Let
&lt;S
lines of curvature being parametric,
be a pseudospherical surface with the linear element (VIII, 22), the and let A be a surface with the same spher
;
ical representation of its lines of curvature as S furthermore, let AI denote the envelope of the plane which makes the constant angle a with the tangent plane at a point of A and meets this plane in a line I/, which forms with the tangent to the curve u = const, at an angle defined by equations (VIII, 35). If MI
M
M
GENERAL EXAMPLES
445
denotes the point of contact of this plane, we drop from MI a perpendicular on L, and NMi, meeting the latter in N. Show that if X and p denote the lengths
MN
they are given by
X
= ( V2? cos
o&gt;
+ V6? sin
o&gt;)
sin
&lt;r,
^
=
(
Vj
sin
w
+ Vt? cos w) sin
o,
where
19.
E and
(?
are the first fundamental coefficients of A.
then
that
Show that when the surface A in Ex. 18 is the pseudospherical surface S, and AI is the Backhand transform Si of S by means of the functions (0, when A is other than S the lines of curvature on the four surfaces S, .4, Si,
&lt;r),
AI correspond, and
the last two have the same spherical representation.
0,
is given all values satisfying equations (VIII, 35) for a given 20. Show that as the locus of the point Jfi, defined in Ex. 18, is a circle whose axis is normal to at M. the surface
A
21.
Show
p.
that
when
A
in Ex. 18
is
a surface of Bianchi of the parabolic type
a.
(Ex. 11,
370) the surfaces
AI
are of the same kind, whatever be
CHAPTER XIV
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
181. Triple system of surfaces associated with a cyclic system. Let S be one of the surfaces orthogonal to a cyclic system, and
1
let its
lines of curvature be
parametric.
The
locus
2
t
of the
circles
which meet S
orthogonally. Hence, by Joachimsthal s theorem ( 59), the line of intersection is a line of curvature for 2 r In like manner, the locus 2 2 of the circles
l
a point
M
l
in the line of curvature v
=
const,
through
is
a surface which cuts
S
u = const, through cuts and the curve of intersection is a line of curva S^ orthogonally, ture on S 2 also. Since the developables of the associated cyclic
which meet S:
in the line of curvature
M
congruence correspond to the lines of curvature on all of the orthogonal surfaces, each of the latter is met by 2 X and 2 2 in a line of curvature of both surfaces. At each point of the circle
through
M the
of curvature v
= const,
tangent to the circle is perpendicular to the line on 2 t through the point and to u = const,
,
Hence the circle is a line of curvature for both 2 X and 2 2 and these surfaces cut one another orthogonally along the circle. Since there is a surface 2 X for each curve v = const, on Sl and a surface 2 2 for each u = const., the circles of a cyclic system and
.
on 2 a
the orthogonal surfaces
may
be looked upon as a system of three
families of surfaces such that through each point in space there passes a surface of each family. Moreover, each of these three sur
faces meets the other
two orthogonally, and each curve
of intersec
( 96) that the confocal quadrics form such a system of surfaces, and another example is afforded by a family of parallel surfaces and
tion
is
a line of curvature on both surfaces.
We
have seen
the developables of the congruence of normals to these surfaces. When three families of surfaces are so constituted that through
each point of space there passes a surface of each family and each
of the three surfaces meets the other
446
two orthogonally, they are
GENERAL EQUATIONS
447
said to form a triply orthogonal system. In the preceding examples the curve of intersection of any two surfaces is a line of curvature
for both.
Dupin showed that
this is a property of all triply
orthog
onal systems.
We
shall prove this
theorem in the next section.
of
182. General equations.
Theorem
Dupin. The simplest exam
ple of an orthogonal system is afforded by the planes parallel to the coordinate planes. The equations of the system are
3
=
1*!,
y
=M
a
,
z
=
i*
8,
where u# u^ u 3 are parameters.
Evidently the values of these parameters corresponding to the planes through a point are the rectangular coordinates of the point. In like manner, the surfaces of each family of any triply orthogonal system may be determined
by a parameter, and the values of the three parameters for the three surfaces through a point constitute the curvilinear coordi nates of the point. Between the latter and the rectangular coor dinates there obtain equations of the form
(1)
x
=/ (w
1
1,
i*
8,
i*,),
y
=/ K,
2
i*
a
,
i*,),
z =/,(!*!,
i*
a
,
i*
8 ),
where the functions
example of
this
is
domain considered. An / afforded by formulas (VII, 8), which define space
are analytic in the
it is
referred to a system of confocal quadrics. In order that the system be orthogonal
necessary and cient that these functions satisfy the three conditions
suffi
v dx
Any
given
dx
^aST
By
ment
_
v dx *to
t
dx
y dx
is
dx
_
when u
t
t
to.
Zto.dut
defined by
(1)
is
one of the surfaces u
this constant value.
= const,
the linear element of space at a point we mean the linear ele at the point of any curve through it. This is
which, in consequence of
(3)
may be written in the 2 = Hl du* + H* du* + HI dui, ds
(2),
parametric form
As thus
assume
defined, the functions
H#
HH
2,
3
are real and
we
shall
that they are positive.
448
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
(3)
From
we have
at once the linear element of
faces of the system.
any of the sur For instance, the linear element of a surface
, 2
U= COnst.
is
rrl J.
Now we shall find that the second quadratic
i
forms of these surfaces
are expressible in terms of the functions and their derivatives. If X^ F., Z denote the directioncosines of the normals to the
H
surfaces u
i
= const, we
have
(5)
,
du.
We
(6)
choose the axes such that
= 41.
the second fundamental coefficients of a sur
In consequence of
face u
{
(5)
const, are defined
by
1
_ ~
where
1
dx d*x
_,
4
,
=
^ dx y
d
2
x_
^u t
_
** H y du
1
i
dx d*x
t
du?
t,
/c,
I
take the values
1, 2,
refers to the
summation
of terms in
3 in cyclic order, and the sign 2 In order #, ?/, 2, as formerly.
differentiate equations (2)
to evaluate these expressions
we
with
respect to
u^ u v u 2
respectively.
This gives
dx
.
_JL__ =
f
o,
dx
dx
^
du.
0.
If
of the three,
each of these equations be subtracted from one half of the sum we have
^
du z du
= o,
0.
dx
d*x
=
du 3 du l
0,
V
= 0;
consequently
D=
THEOREM OF DUPIN. EQUATIONS OF LAME
If the
first
449
and third
u z and u s respectively, respect to u v we have
dx d*z
of (2) be differentiated with respect to and the second and third of (4) with
dx
tfx
~~
2 2
a#
~
2
du^
dx tfx
y
J5T
t
dx
tfx
3H
3
S
Hence we have
Z&gt;
Proceeding in like manner, we find the expressions for the other s, which we write as follows
:
3 in cyclic order. From the sec ond of these equations and the fact that the parametric system on each surface is orthogonal, follows the theorem of Dupin
where
i,
K,
I
take the values
1, 2,
:
The surfaces of a
lines
triply orthogonal system meet one another in
of curvature of each.
Lame".
183. Equations of
conditions to be satisfied
by Hv H^ H
By means
of these results
z,
we
find the
in order that (3)
may
be
the linear element of space referred to a triply orthogonal system of surfaces. For each surface the Codazzi and Gauss equations
must be
satisfied.
When
the above values are substituted in these
equations,
we
and sufficient that the functions H satisfy
find the following six equations
:
which
it is
necessary
PH
/ox
{
1
dH^H,
1
~ gJgjgjr^
H
t,
/^,
where
I
take the values
Lame",
1, 2, 3
in cyclic order.
These are
first
the equations of
being named
for the
geometer who
deduced them.*
*
Lemons sur
les
coordonntes curvilignes
et leurs diverses applications, pp. 7379.
Paris, 1859.
450
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
of the surfaces there
For each
form (V, 16).
equations
When
is a system of equations of the the values from (7) are substituted in these
we have
&lt;
t
du,
Recalling the results of
65,
we have
that each set of solutions
of equations (8), (9) determine a triply orthogonal system, unique to within a motion in space. In order to obtain the coordinates of space referred to this system, we must find nine functions
JQ,
r;.,
Zf
which
satisfy (10)
and
1,
2^=0.
+ H^X du + H X du
Z 2 Z Z
(=*=*)
Then
the coordinates of space are given by quadratures of the form
x
=
I
H^XI du
l
s
.
denotes the principal radius of a surface u = const, in the direction of the curve of parameter U K we have, from (7),
If p. K
f
,
rm
l
1Pi.
eter
Let p denote the radius of first curvature of a curve of param u r In accordance with 49 we let w 1 and w[ ?r/2 denote the
angles which the tangents to the curves of parameter u 3 and u 2 respectively through the given point make, in the positive sense, with the positive direction of the principal normal of the curve
of parameter
/i
ur
Hence, by (IV,
16),
we have
o\
Pi
Pn
Pi
Pzi
these equations and similar ones for curves of parameter u and u s we deduce the relations
From
,
(13)
1 = 1 + 1,
Pi
Pl&gt;
tan5 = 6t,
(
Pfi
P
ONE FAMILY OP SURFACES OF REVOLUTION
where
2, /c, I
451
take the values
1, 2, 3 in cyclic order.
it
Moreover, since
the parametric curves are lines of curvature, that the torsion of a curve of parameter u is
i
follows from
(59)
(14)
ll^i.
r
{
Hi
du,
184. Triple systems containing one family of surfaces of revolution.
of plane curves and their orthogonal trajectories the plane be revolved about a line of the plane as an axis, the two families of surfaces of revolution thus generated, and the planes
Given a family
;
if
through the axis, form a triply orthogonal system. We inquire whether there are any other triple systems containing a family of
surfaces of revolution.
Suppose that the surfaces u s = const, of a triple system are sur faces of revolution, and that the curves u 2 = const, upon them are
the meridians.
Since the latter are geodesies,
we must have
From
(8) it
follows that either
dH. i
du s
=
n
0,
or
dffs
8
n = 0.
du 2
follows from (11) that l//o 31 = 0. Consequently, the surfaces of revolution w 3 = const, are developables, that is, either circular cylinders or circular cones. Furthermore, from (15) and
In the
first
case
it
(11),
we have
l//o 21
=0,
so that the surfaces
u2
= const,
also are
developables, and
in addition
we
have, from (13), that l//^
= 0,
that
lines and consequently is, the curves of parameter u^ are straight the surfaces u^= const, are parallel. The latter are planes when the surfaces u s = const, are cylinders, and surfaces with circular lines of curvature
when u s =
const, are circular cones. Conversely,
(
132, it follows 187) and from with parallel generators, or that any system of circular cylinders locus any family of circular cones whose axes are tangent to the of the vertex, leads to a triple system of the kind sought. consider now the second case, namely
from the theorem of Darboux
We
452
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
(11)
From
we
it
surfaces w a =
const, are planes.
meridians,
= 0; consequently the l//o 28 Since these are the planes of the follows that the axes of the surfaces coincide, and
find that l//o 21
=0, and
consequently the case cited at the beginning of this section only one for nondevelopable surfaces.
is
the
it
In 119 185. Triple systems of Bianchi and of Weingarten. was found that all the Bianchi transforms of a given pseudo
of the same total spherical surface are pseudospherical surfaces curvature, and that they are the orthogonal surfaces of a cyclic
system of
circles
circles of constant radius.
Hence the
totality of these
and surfaces constitutes a
were
first
triply orthogonal system, such
that the surfaces in one family are pseudospherical.
of this sort
As systems
119), they
considered by Ribaucour
(cf.
are called the triple si/stems of Ribaucour. proceed to the consideration of all triple systems such that the surfaces of one
We
family are pseudospherical.
*
These systems were
first
studied by
Bianchi, and consequently Darboux f has called them the systems of Bianchi.
From
119
it
ture of a pseudospherical surface of curvature chosen that the linear element takes the form
(1 6)
follows that the parameters of the lines of curva a l/a can be so
d
o&gt;
a
= cos
a
o)
du* + sin
a
o&gt;
dv\
where
is
a solution of the equation
d
a
o&gt;
d
a
o&gt;
_ sin
o&gt;
cos
a)
In this case the principal radii are given by
1
tan
&)
1
cot
a)
(18)
Pl
a
p,
a
In general the total curvature of the pseudospherical surfaces of a system of Bianchi varies with the surfaces. If the surfaces w = const, are the pseudospherical surfaces, we may write the
3
curvature in the form
Annali, 8er.
Vol.
t II,
7&gt;vow
1/f^, where
(188,",),
?/8 is
a function of
?/
alone.
2,
Vol. XIII
pp. 177234; Vol.
et les
XIV
(1880), pp. 115130; Lezioni,
chap, xxvii.
wr
les ni/stemes
orthogonaux
coonlonntes curvilignes, pp. 308323.
Paris, 18U8.
TRIFLE SYSTEMS OF BIANCHI
In accordance with (11) and (18)
_1 P* ,
453
we put
tan&lt;w
1
dff,
(19)
1
g//
cot
&lt;a
s)
tf

?)
ff
=
If
these values of

and
be substituted in equations
(8)
for*
r/r,
(K,,
equal to 1 and 2 respectively,
1
we
,
obtain
1
BJf, l
=
3to
tan
dlfa
=
o&gt;
=
do)
cot
a)
From
(20)
these equations
we
&lt;
have, by integration,
cos
co,
ffl
=
13
H=
z
&lt;/&gt;.,
3
sin
CD,
where $ 13 and $ 23 are functions independent of w a and u respectively. \Vo shall show that both of them are independent of u 3
v
.
When
we have
the values of
respectively
jff
H
l
and !! from
to
to
(20) are substituted in (19),
= fr
= fr
it
/.
C ot
cw
(
tan
3 log
&lt;.
I
(21)
/
//
tan
o&gt;
[
cot
o&gt;
\
A d da)   log z ,5w, ^3
\
\
!
/
From
these equations
follows that
Hence, unless
l;l
and
^&gt;.,
;l
the ratio of a function of u
are independent of w 3 , tan is equal to and w 3 and of a function of w 2 and 3 v
o&gt;
?&lt;
.
consider the latter case and study for the moment a partic ular surface i/ 3 = c. By the change of parameters
(^..(MP
We
cjau^
the linear element of the surface reduces to (16), and (22) becomes
tan
co
=
v respectively.
where
f and V
is
are functions of
u and
obtain
V
a
,
When
this
substituted in (17),
we
w^Ox
/C\^,,,
iv
i
n
,
*S
v
I
\
* wrft
.
(u"T
454
TKIPLY ORTHOGONAL SYSTEMS OF SURFACES
with respect to u
If this equation be differentiated successively
and
v,
we
find
/U"\
f
1
r
\u)
unless
~uu~
+
/V"\
1
=
\r)
~vv
this it follows that
V or
V
is
equal to zero.
From
where K denotes a constant. Integrating, we have
U"=2icU*+aU,
V"=2tcV
s
+
l
a and
y3
being constants, and another integration gives
U
*=KU*+a(r*+&lt;y,
F
2
=*F + /3F
4
2
+S.
find
When
these expressions are substituted in (23),
we
This condition can be
Hence
alone.
U
satisfied only when the curvature is zero. be zero, that is, &) must be a function of u or v In this case the surface is a surface of revolution. In accord
1
or
V must
ance with
184 a
triple
system of Bianchi arises from an
infinity
of pseudospherical surfaces of revolution with the
same
axis.
(f&gt;
When
exception
is
made
of this case, the functions
.
13
and
&lt;/&gt;
23
in (20) are
independent of u s
Hence
the parameters of the sys
tems may be chosen so that we have
/2^x
H
IT
H
f7
When
~
these values are substituted in the six equations
(8), (9),
they reduce to the four equations
2
&)
2
ft)
sin
to
cos
&)
=
duf
0M*
cot
U,
III
ft)

f
tan
to
= 0,
1
2
g&lt;
(25)
0/1
cu v \cos
_d_ /
ft)
g
2
ft)
\
1
d
3
/sinftA
ft)
du 1 duj
d
2
U
1
du 3 \
d
U
=()
G)
3
/
ft)\
sin 1
du 2 du 2 du 3
1
&)
&)
\
/cos
do
ft)
d^co
_
du 2 \sin
du 2 duj
U
3
du 3 \
U
Q
3
/
cos
cu v du l du z
TRIPLE SYSTEMS OP WEINGAKTEN
455
Darboux has inquired into the generality of the solution of this system of equations, and he has found that the general solution
involves five arbitrary functions of a single variable. shall not give a proof of this fact, but refer the reader to the investi
We
gation of Darboux.* turn to the consideration of the particular case where the total curvature of all the pseudospherical surfaces is the same, 1 without any loss of generality. which may be taken to be
We
As we
triple
systems of this sort were
first
discussed by Weingarten,
follow Bianchi in calling them systems of Weingarten. kind are the triple systems of Ribaucour.
Of
this
For
(26)
this case
we have U3 = 1,
2
so that the linear element of space
is
ds
= cos
2
&)
2 dul + sin a) du*
Since the second of equations (25)
the forms
c
may
\ cos
ft)
be written in either of
a
/
\
n
ft)
ff
w
\
cw
c
z
a)
,
du 2 du s /
)=
du 2 du^ du s
du 2 \cos
if
ft)
du^ du 3 /
2
sin
/
j
ft)
du l du 2 du s
o2
We pUt
/
j
\2
\2
/o
ycos
it
ft)
du 1 du^l
V
sin
&)
du z cu z j
\^
3
follows from the last two of (25) and from (27) that
eter
is a function of u s alone. Hence u z an operation which will not
&lt;J&gt;
But by changing the param
affect the
,
form of
(26),
we can
give
(28 )
&lt;&
a constant value, say
c.
\cos
 Bu du
ft)
+ =
l
Consequently we have
c.
3
J
\sm co du 2 du 3 /
\^
3
/
Bianchi has shown f that equation (28) and the
first of (25)
are
= Consequently the equivalent to the system (25), when Z78 l. of the determination of triple systems of Weingarten is problem the problem of finding common solutions of these two equations.
*L.c., pp. 313, 314; Bianchi, Vol. II, pp. 531, 532.
t Vol. II, p. 550.
456
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
EXAMPLES
Show
that the equations
1.
x
=
r cos
u cos v,
y
=
r cos
u
sin u,
z =: r sin
w
define space referred to a triply orthogonal system.
2. A necessary and sufficient condition that the surfaces u s = const, of a triply What are the s be a function of u 3 alone. orthogonal system be parallel is that other surfaces u\ = const., u% = const.?
H
3. Two nearby surfaces u s = const, intercept equal segments on those orthog onal trajectories of the surfaces w 3 const, which pass through a curve s = const. on the former; on this account the curves #3 = const, on the surfaces u 3 = const.
H
are called curves of equidistance.
4.
Let the surfaces w 3
=
const, of a triple
system be different positions of the
in the direction of its axis.
same pseudosphere, obtained by translating the surface
Determine the character of the other surfaces of the system.
5.
Derive the following results for a triple system of Weingarten
:
C+ V/8w\2
U
where the
surface u 3
differential
parameter
is
formed with respect
to the linear element of a
=
const.,
and
p g is the radius of geodesic
curvature of a curve
=
w3
const
const.
on
that the curves of equidistance on the surfaces u s are geodesic parallels of constant geodesic curvature.
this surface.
6.
Show
=
Show
that
when
c in (28) is
is
curves of parameter u 3
to (12)
equal to zero, the first curvature l/p 8 of the constant and equal to unity; that equations similar
a2
,
become
2
=

sin
o&gt;
cos w 3
last
_
8&lt;a
u&gt;
=
dot
cos u sin
u&gt;
3
;
that
if
we put
cd
w 3 the
,
two of equations
dO
^H
(25),
where
U=
&
;
1,
may
be
written
gw   = sm 6 cos w,
1
aw  =
cos
sin
u
and that
&d  =
8u
^
Sin 6 COS
0,
/
/
cos6
is
 1
8*6
\
2
J
f /
/
1
8*6
\
]
2
= /de\*
/
)

.
When
c
=
in (28) the
system
said to be of constant curvature.
7. A necessary and sufficient condition that the curves of parameter u s of a system of Weingarten be circles is that w 3 be independent of u 3 In this case (cf. Ex. 6) the surfaces u s = const, are the Bianchi transforms of the pseudospherical surface with the linear element
.
ds*
=
cos*0du*
THEOREM OF RIBAUCOUR
Theorem Ribaucour *
186.
:
457
is
of Ribaucour.
The following theorem
due to
Griven a family of surfaces of a triply orthogonal system
and
their
orthogonal trajectories; the osculating circles to the latter at their
points of meeting with any surface of the family form a cyclic system.
In proving this theorem we
fied
first
derive the conditions to be satis
by a system of circles orthogonal to a surface S so that they may form a cyclic system. Let the lines of curvature on S be parametric
and refer the surface
to the
moving
trihedral
whose
x
and
?/axes
are tangent to the curves v
(29)
If
&lt;
= const.,
u
=
const.
We
have (V, 63)
^=n=p = qi =0.
P
denotes the angle which the plane of the circle through a the angle which point makes with the corresponding zzplane, the radius to a point of the circle makes with its projection in
the z^plane, and the radius of the circle, the coordinates of with reference to the moving axes are
R
P
x
= R(\ f cos 0) cos $,
sin 6 cos
y
=^(1+ cos 0) sine/),
sin 6 sin $,
z=lism0.
circle at
6.
Moreover, the directioncosines of the tangent to the
&lt;,
P are
cos
If
we express
the condition that every displacement of
P
must be
at right angles to this line,
we
have, from (29) and (V, 51),
dB
 [sin B( [_
 f sin 6
.
\R du J\ dR \jri
+ lL^i)+ q cos 0(1 + cos 0)1 du R/ J
[_
cv
+ A__ K
77,
sin&lt;f&gt;\
_p
sm 0(i +
."I
cos B)\dv
J
is
7
=
0.
/
The
condition that this equation admit an integral
reducible to
cosjAI
[sin
L
4,
cose/)
E
/J
R
Hence, as remarked before
(
174),
if
there are three surfaces orthog
is cyclic.
onal to a system of circles, the system
*
Comptes Rendus, Vol.
LXX
(1870), pp. 330333.
458
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
that
(f&gt;\
The condition
d_ /i
?1
it
be cyclic
d_
is
sin
cu \
(30)
R
cos
(/
/
dv\
R
,
sin
&lt;f&gt;
d /sin
(f)
\
d
icos&lt;f)
\_
/&gt;
Since the principal radii of
(31)
S
are given
by
i=
.
i=
J.
S
:
the second of equations (30) reduces to the first when or a plane. Hence we have incidentally the theorem
is
a sphere
A
any
twoparameter system of
circles
orthogonal
to
a sphere and
to
other surface constitute a cyclic system.
return to the proof of the theorem of Ribaucour and apply the foregoing results to the system of osculating circles of the
We
curves of parameter u 3 of an orthogonal system at their points of
intersection with a surface
u= const.
o
From
equations similar to (12)
cos
&lt;j)
we
have, by (11),
1
&lt;f&gt;
1
d//.,
sin
dH
z
and the equations analogous
1
to (31) are
1
/&gt;,
q
1
211^
Z
_
2
p
l
_
1
3H
Z
pn
^
H^H
du 3
//2
7/2 // 3 du s
these values are substituted in equations (30) the first vanishes identically, likewise the second, in consequence of equa tions (8). Hence the theorem of Ribaucour is proved.*
naturally arises whether any family of surfaces whatever forms part of a triply orthogonal system. This question will be answered with the aid
187.
of
When
Theorems
Darboux.
The question
of the following theorem of Darboux, f
which we establish by
his
methods
:
necessary and sufficient condition that two families of surfaces orthogonal to one another admit of a third family orthogonal to both is that the first two meet one another in lines of curvature.
* For a geometrical proof the reader
is
A
referred to Darboux,
I.e., p.
77.
t L.c.,
pp. 68.
THEOREM OF DARBOUX
Let the two families of surfaces be defined by
(32)
459
a(x, y,
b are the
z)
= a,
@(x, y,
z)
= b,
is
where a and
parameters.
The
condition of orthogonality
dx
~dx
dy^y
~dz~dz~
In order that a third family of surfaces exist orthogonal to the
surfaces of the other families, there
satisfying the equations
.
must be a function
7(2, y, z)
,
_
dz
_
~dx
dx dx
dydy
dz~
dx
~d^
~dz~dz~
If dx, dy, dz
ment
of a point
denote the projections on the axes of a displace on one of the surfaces 7 = const., we must have
dx
da
dx
dy
da dy
dz
da
dz
= 0.
Idx
~dy
~dz
that
This equation is of the form (XIII, 9). The condition (XIII, 10) it admit of an integral involving a parameter is
da
dx
dz*
dz dxdz
dx
dz*
dz dxdz
~^ydxdy~~dx^y* dydxdy ^x ty*\~ where S indicates the sum of the three terms obtained by permut ing x, y, z in this expression. If we add to this equation the identity
d (a,
/3) 01* \da
+
{
^
^\
v
f~f
u ff ^\
^
i/
LV
~i
i
f*
=~~
\
5
the resulting equation
may
be written in the form
d/3\
da T~ dx da
(34)
dp T~ dx
dj3
Ja
\
^~ dx
,/ p ~ ^l&
da
= 0,
S( I
ill
r\
I
da
dz
d/3
)_ gr/3 *1
I
_
r\
.
dz
460
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
we have introduced
the symbol
where, for the sake of brevity, defined by 8(0,
4&gt;),
equation (33) be differentiated with respect to be written
If
x,
the result
may
Consequently equation (34)
da
dx
r\
is
reducible to
df3
dx
ccc
cp
dy
O
/O
(35)
=o,
da
dz
dz
which
is
therefore the condition
upon a and
/S
in order that the
desired function 7 exist. a = const, displacement along a curve orthogonal to the surfaces
A
is
given by
^ = ^_^. da da
da
dx
dy
dz
/3
Such a curve
it satisfies
lies
upon
a surface
=
const, and since, by (35),
the condition
= 0,
it is
a line of curvature on the surface
(cf.
Ex.
3, p.
247).
fi
Hence
the curves of intersection of the surfaces a
= const.,
= const.,
are lines of being the orthogonal trajectories of the above curves, = const. And by Joachimsthal s theo curvature on the surfaces ft rem ( 59) they are lines of curvature on the surfaces a = const,
also.
Having thus established the theorem
answer the question
at the
of
Darboux, we are in
a position to
beginning of this section.
TRANSFORMATION OF COMBESCURE
;
461
Given a family of surfaces a const. the lines of curvature in one family form a congruence of curves which must admit a family of orthogonal surfaces, if the surfaces a = const, are to form part
of an orthogonal system. If this condition is satisfied, then, accord ing to the theorem of Darboux, there is a third family of surfaces
which together with the other two form an orthogonal system.
If
Xv
Yv Z
1
lines of curvature in
denote the directioncosines of the tangents to the one family on the surfaces a const., the ana
be a family of surfaces orthogonal to
that the equation
lytical condition that there
these curves
is
admit an integral involving a parameter. The condition for
this is
In order to find
X^ Y^ Z we remark
1
that since they are the direc
tioncosines of the tangents to a line of curvature
we must have
and similar equations in
Y, Z,
where the function X
is
a factor of
proportionality to be determined arid Jf, Y, Z are the directioncosines of the normal to the surface a const. Hence, if the
=
surfaces are defined
by a
= const.,
the functions
Xv
Y^ Z^
of a,
are
expressible in terms of the first
and second derivatives
and
so equation (36) is of the third order in these derivatives. fore we have the theorem of Darboux*:
There
The determination of all triply orthogonal systems requires the integration of a partial differential equation of the third order.
Darboux has given the name family of Lame to a family of surfaces which forms part of a triply orthogonal system.
188. Transformation of Combescure.
We close our study of triply
orthogonal surfaces with an exposition of the transformation of Combescure,^ by means of which from a given orthogonal system
others can be obtained such that the normals to the surfaces of
one system are parallel to the normals to the corresponding sur
faces of the other system at corresponding points.
*
L.c., p. 12.
f
Annales de
I
Ecole Normale Superieure, Vol. IV (1867), pp. 102122.
462
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
make use
of a set of functions
/3iK ,
We
introduced by Darin space of
boux * in his development of a similar transformation n dimensions. By definition
In terms of these functions equations
the form
(8),
(9) are expressible in
37
&lt; &gt;
$* t
(10)
+
and formulas
(38)
become
A.jr.arr,,
Equations
(37), (38) are the
the expression
^^ ^^ ^^
+
+
From
their
necessary and sufficient conditions that
we have another set of functions H[, 772 //8 satisfying the six conditions
it is
be an exact differential.
form
seen that
if
,
39
&lt;
&gt;
*/3tK
where the functions
have the same values as for the given
system, the expression
XJI[ dUl f JT2 //2 du 2 + Xfi du a
and similar ones
tures
in F, Z, are exact differentials,
,
and so by quadra
desired property.
we obtain an orthogonal system possessing the
In order to ascertain the analytical character of this problem, we eliminate H[ and H^ from equations (39) and obtain the three
equations
n
_
.._
,
^
du,
cu^
.
The general integral of a system of equations of this kind involves three arbitrary functions each of a single parameter u When one
t
*L.c., p. 161.
GENERAL EXAMPLES
has an integral, the corresponding values of by (39). Hence we have the theorem
:
463
H^ H^ are given directly
With every triply orthogonal system there is associated an infinity of others, depending upon three arbitrary functions, such that the normals to the surfaces of any two systems at corresponding points
are parallel.*
1.
EXAMPLES
In every system of Weingarten for which c in (28) is zero, the system of cir cles osculating the curves of parameter u s at points of a surface w 3 = const, form a
system of Ribaucour
2.
(
185).
orthogonal trajectories of a family of Lame" are twisted curves of the same constant first curvature, the surfaces of the family are pseudospherical
If the
surfaces of equal curvature.
3. Every triply orthogonal system which is derived from a cyclic system by a transformation of Combescure possesses one family of plane orthogonal trajectories.
4.
If the
system of circles osculating these trajectories family may be obtained from the given system 5. Determine the triply orthogonal systems the transformation of Combescure to a system
orthogonal trajectories of a family of Lame* are plane curves, the cyclic at the points of any surface of the
by a transformation of Combescure. which result from the application of of Ribaucour ( 185).
GENERAL EXAMPLES
1.
If
an inversion by reciprocal
radii
(
80) be effected
upon a
triply orthogonal
system, the resulting system will be of the same kind.
2. Determine the character of the surfaces of the system obtained by an inversion from the system of Ex. 1, 185, and show that all the curves of intersection are circles. 3.
Establish the existence of a triply orthogonal system of spheres.
4. necessary and sufficient condition that the asymptotic lines correspond on the surfaces u% = const, of a triply orthogonal system is that there exist a relation
A
of the
form
0j,
2
&lt;t&gt;s
03
are functions independent of w 3
is satisfied,
.
=
?
where
5.
When
us
the condition of Ex. 4
those orthogonal trajectories of the
surfaces u s
face
=
const,
=
const,
asymptotic lines
6.
which pass through points of an asymptotic line on a sur constitute a surface S which meets the surfaces u s = const, in of the latter and geodesies on
&lt;S.
that the asymptotic lines correspond on the pseudospherical surfaces of a triple system of Bianchi. 7. Show that there exist triply orthogonal systems for which the surfaces in one
Show
family, say u$
const., are spherical,
and that the parameters can be chosen so that
sinh
6,
HI =
Find the equations of
8.
cosh
8,
Hz =
H
3
= US
CUz
is
.
Lame"
for this case.
Every oneparameter family of spheres or planes
*Cf. Bianchi, Vol.
II, p.
a family of
Lame
.
494.
464
TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
9. In order to obtain the most general triply orthogonal system for which the surfaces in one family are planes, one need construct an orthogonal system of curves in a plane and allow the latter to roll over a developable surface, in which
case the curves generate the other surfaces. When the developable determination of the system reduces to quadratures.
10.
of
is
given, the
Show
that the most general triply orthogonal system for which one family
Lame"
consists of spheres passing through a point can be found
by quadratures.
.
11.
Show Show
that a family of parallel surfaces
is
a family of
Lame
3
that the triply orthogonal systems for which the curves of parameter are circles passing through a point can be found without quadrature.
12.
13. By means of Ex. 6, 185, show that for a system of Weingarten of constant curvature the principal normals to the curves of parameter w 3 at the points of meet ing with a surface u 3 = const, form a normal pseudospherical congruence, and that the surfaces complementary to the surfaces w 3 = const, and their orthogonal tra
jectories constitute a system of
14.
By means
of Ex. 13
Weingarten show that for a
of constant curvature.
triple
system arising from a system of
of
Weingarten
of constant curvature
by a transformation
Combescure the osculat
ing planes of the curves w 3 = const., at points of a surface u s = const., envelop a surface S of the same kind as this surface M 3 = const. ; and these surfaces S and
their orthogonal trajectories constitute a system of the
same kind
as the one result
ing from the Combescure transformation of the given system of Weingarten.
15.
Show
system of Bianchi be plane
d(*t
that a necessary condition that the curves of parameter u\ of a triple is that w satisfy also the conditions
=
023
sm w,
.
Set
=
0i 3 sin w,
(cf.
di/2
dui
where
23
Show
that
and 0i 8 are independent of HI and w 2 respectively if 0is and 02 3 satisfy the conditions
Ex.
5, p. 317).
where a and
given by
b are constants
and U&
is
an arbitrary function of w 8 the function w, g0 23 a0i 8
,
COS
W
=
determines a triply orthogonal system of Bianchi of the kind sought.
16.
(25)
When
Z73
=
1
and w
is
independent of u 2 the
,
first
and fourth of equations
may
be replaced by
gw
=
sin w.
dui
Show
that for a value of
satisfying this condition
/
(
the expressions
HI =
cos w
rfadu s
I
\J sin w

\
+
_(
0i
)
 r^cosu, 3 du
/ J
:
/
sin
w
and the other equations
f 0i,
(25)
H
z
=
sin
w
(
\J
(
(
C^J^ sin w
I
0A Jr0 3 cZM8 +
/
2
,
HB ~
\J
ffoduz sin w
+
0i
\ du
/
5u 3
GENERAL .EXAMPLES
3 are functions of MI, u 2j 2 0i, differentiation, define a triply orthogonal
465
where
,
w 3 respectively, and the accent indicates system for which the surfaces M 3 = const,
are molding surfaces.
17.
Under what conditions do the functions
d2
sin
w
Z72 and Z7s are functions of M 2 and M 3 respectively, determine a triply orthog onal system arising from a triple system of Bianchi by a transformation of Combescure ? Show that in this case the surfaces w 2 = const, are spheres of radius Z72 and
where
,
that the curves of parameter M 2 in the system of Bianchi are plane or spherical.
18.
Prove that the equations
y
Z
B(UI
b) C)
m
i(u 2
i(lt 2
b) C)
m
*(u s
2(w 3
6)
c)
m
s,
"3,
=

C(Ui
wl
w
are constants, define space referred to a triple system of surfaces, such that each surface is cut by the surfaces of the other two families in
a, 6, c,
t
where A, B, C,
m
a conjugate system.
19. Given a surface 8 and a sphere S; the circles orthogonal to both constitute a cyclic system hence the locus of a point upon these circles which is in constant crossratio with the points of intersection with S and S is a surface Si orthogonal to the circles Si may be looked upon as derived from S by a contact transformation which preserves lines of curvature such a transformation preserves planes and
;
;
;
spheres.
20.
When S
all
of Ex. 19
surface which
is
is a cyclide of Dupin, so are the surfaces Si, and also the the locus of the circles which meet S in any line of curvature
;
hence
of these surfaces
form a
Z7,
triple
system of cyclides of Dupin.
21. Given three functions
defined by
Ui =
imuf
+ 2 mm + p^

(i
=
1, 2, 3)
where
m
t ,
Wj,
pi are constants satisfying the conditions
Sm = 0,

t
Sn =
t
0,
Sp

t
=
;
and given
also the function
N=
where
,,
tti(M 2

u 3 )VUi
f
a 2 (M 8 
MI)
VU^ +
&lt;*
3
(MI

+ pZniiUi +
0,
7 (PiMaWa
+ PZ^UI
7 are constants determine under what condition the functions U S  Ui W 2  US MI Ma TT
&l

;
;=
N^lfi
determine a triply orthogonal system. and that they are cyclides of Dupin.
22. Determine
N^U
7=
^33=
2
p= NVU
3
Show
that
all
of the surfaces are isothermic,
whether there exist triply orthogonal systems of minimal surfaces.
INDEX
The numbers refer to pages. References to an author and his contributions are made in the form of the first Bianchi paragraph, whereas when a proper name is part of a title the reference is given the form as in the second Bianchi paragraph.
Acceleration, 15, 60
spherical
lines
of
curvature),
;
;
315
;
Angle between curves, 74, 200 Angle of geodesic contingence, 212
Applicable surfaces, definition, 100 to the plane, 101, 156 invariance of invariance geodesic curvature, 135
; ; ;
(associate surfaces), 378 (cyclic con gruences of Ribaucour), 435 (cyclic
of total curvature, 156 solution of the problem of determining whether
;
systems), 441 Bianchi, transformation of, 280283, 290, 318, 320, 370, 456 surfaces of, 370, 371, 442, 443, 445 generalized
;
;
two given surfaces are applicable, 321326 pairs of, derived from a given pair, 349. See Deformation of
;
surfaces
Area, element of, 75, 145 Area of a portion of a surface, 145, 250 minimum, 222 Associate surfaces, definition, 378 de termination, 378381; of a ruled surface, 381; of the sphere, 381; ap plicable, 381; of the right helicoid, 381 of an isothermic surface, 388 of of pseudospherical surfaces, 390 characteristic .quadrics, 390, 391; property, 425 Asymptotic directions, definition, \2S *Asymptotic lines, definition, 128 para metric, 129, 189194 orthogonal, 129 spherical represen straight, 140, 234 tation, 144, 191193; preserved by protective transformation, 202 pre served in a deformation, 342347
; ; ;
;
transformation of, 439 triply orthog onal systems of, 452454, 464, 465 Binormal to a curve, definition, 12 spherical indicatrix, 50 Bmormals which are the principal nor mals to another curve, 51
;
;
Bonnet (formula of geodesic curvature),
(surfaces of constant curvature), 179; (lines of curvature of Liouville type), 232 (ruled surf aces), 248 (sur faces of constant mean curvature),
; ;
136
;
298
IJour (helicoids), 147; (associate isother mic surfaces), 388
;
Canal surfaces, definition, 68
of center, 186 Catenoid, definition, 150
;
;
surfaces
;
;
;
;
;
Backhand, transformation of, 284290 Beltrami (differential parameters), 88,
90; (geodesic curvature), 183; (ruled Wsurfaces), 299 (applicable ruled surfaces), 345 (normal congruences),
; ;
adjoint sur 267 surfaces applicable to, 318 Cauchy, problem of, 265, 335 Central point, 243 Central plane, 244 Cesaro (moving trihedral), 8S Characteristic equation, 375 Characteristic function,, 374, 377 face
of,
;
Characteristic lines, 13Q, 131
ric,
;
paramet
203
;
Characteristics, of a fanUly of surfaces, 5961 of the tangent pfones to a sur/
face, 126 Christoffel (associate isothermic sur faces), 388 Christoffel symbols, definition, 152, 153 ; relations between, for a surface and
its
403 Bertrand curves, definition, 39 proper parametric equations, 51 ties, 3941 on a ruled surface, 250 ; deformation, 348 Bianchi (theorem of permutability), 286288 (surfaces with circular lines of curvature), 311; (surfaces with
; ;
;
spherical representation, 162, 193,
;
;
201
Circle, of curvature, 14
osculating, 14
*
surfaces are listed under the latter.
References to asymptotic
lines, geodesies, lines of
curvature,
etc.,
on particular kinds of
467
468
INDEX
;
Circles, orthogonal system of, in the plane, 80, 97 on the sphere, 301 Circular lines of curvature, 149, 310,
Correspondence with orthogonality of linear elements, 374377, 390 Corresponding conjugate systems, 130
Cosserat (infinitesimal deformation), 380, 385 Crossratio, of four solutions of a Riccati equation, 26 of points of intersection of fourcurved asymptotic lines on a ruled surface, 249 of the points in which four surfaces orthogonal to a cyclic system meet the circles, 429 Cubic, twisted, 4, 8, 11, 12, 15, 269 Curvature, first, of a curve, 9; radius of, 9; center of, 14; circle of, 14; constant, 22, 38, 51 Curvature, Gaussian, 123 geodesic (see
; ; ;
316, 423, 446 Circular point on a surface, 124 Codazzi, equations of, 155157, 161, 168,
170, 189, 200
Combescure transformation, of curves,
of triple systems, 401465 surface, 184, 185, 283, 290, 370, 464 Conforinal representation, of two sur faces, 98100, 391; of a surface and its spherical representation, 143 of a surface upon itself, 101103 of a plane upon itself, 104, 112 of a sphere upon the plane, 109 of a sphere upon itself, 110, 111; of a pseudospherical surface upon the plane, 317 Conformalcon jugate representation of
;
50
Complementary
;
;
;
;
Geodesic) Curvature, mean, of a surface, 123, 126, 145 surfaces of constant (see Sur
;
face)
two surfaces, 224 Congruence of curves, 426 normal, 430 Congruence of straight lines (rectilinear),
;
definition, 392 normal, 393, 398, 401, 402, 403, 412, 422, 423, 437; associate
;
normal, 401403, 411; ruled surfaces, 393, 398, 401 limit points, 396 prin cipal surfaces, 396398, 408 principal
; ; ;
Curvature, normal, of a surface, radius of, 118, 120, 130, 131, 150; principal radii of, 119, 120, 291, 450 center of, 118, 150; principal centers of, 122 Curvature, second, of a curve, 16 con stant, 50. See Torsion Curvature, total, of a surface, 123, 126,
;
;
planes, 396, 397; developable*, 398, 409, 414, 421, 432, 437; focal points, 398, 399, 425; middle point, 399; middle surface, 399, 401, 408, 413, 421424 middle envelope, 413, 415 focal planes, 400, 401, 409, 416 focal
;
211
145, 155, 156, 160, 172, 186, 194, 208, radius of, 189 surfaces of con
; ;
;
;
stant (see Surface) Curve, definition, 2; of constant first curvature, 22, 38, 51; of constant form of a, 18 torsion, 50 Cyclic congruences. See Congruences
;
surfaces, 400, 406, 409411, 412, 414, 416, 420 derived, 403405, 411, 412 isotropic, 412, 413, 416; of Guichard, 414,415,417,422,442 pseudospherical, 184, 415, 416, 464 W, 417420, 422, 424 of Ribaucour, 420422, 424, 425, 435, 442, 443 mean ruled surfaces, 422, 423, 425 cyclic, 431445 spher ical representation of cyclic, 432433 cyclic of Ribaucour, 435, 442, 443 developables of cyclic, 437, 441
;
;
;
;
;
;
;
;
Cyclic system, 426445 definition, 426 of equal circles, 430, 443 surfaces orthogonal to, 436, 437, 444, 457; planes envelop a curve, 439, 440 planes through a point, 440, 441 planes depend on one parameter, 442 triple system associated with a, 446 associated with a, triple system, 457;
; ;
;
;
;
;
;
;
458
Cyclides of Dupin, 188, 312314, 412, 422, 465
;
;
normal cyclic, 437 Conjugate directions, 126, 173
radii in, 131
;
normal
;
D, Z7,
Conjugate system, definition, 127, 223 parametric, 195, 203, 223, 224 spher ical representation of, 200 of plane curves, 224 preserved by projective 202 preserved in a transformation, deformation, 338342, 348, 349 Conjugate systems in correspondence, 130 Conoid, right, 56, 58, 59, 68, 82, 98, 112,
; ;
;
A Darboux
170 191 195
;
for the definition, 115 ing trihedral, 174 definition, 386 Jb i
7)",
;
&"&gt;
mov
(moving trihedral), 168,
169,
;
(asymptotic lines parametric), (conjugate lines parametric), (lines of curvature preserved by an inversion), 196 (asymptotic lines
;
;
;
and conjugate systems preserved by
projective transformation), 202 (geo desic parallels), 216, 217 (genera tion of new surfaces of Weingarten), 298 (generation of surfaces with plane lines of curvature in both sys 304 ; (general problem of tems),
; ;
120, 195, 347
Coordinates, curvilinear, on a surface,
55 curvilinear, in space, 447 sym metric, 9193 tangential, 163, 194,
;
;
;
;
201;
elliptic,
227
INDEX
deformation), 332 (surfaces appli cable to paraboloids), 367 (triply
;
469
;
;
orthogonal systems), 458461 Darboux, twelve surfaces of, 391; de rived congruences of, 404, 405 Deformation of surfaces (see Applicable
of surfaces of revolution Surfaces of revolution) of mini mal surfaces, 264, 269, 327330 of surfaces of constant curvature, 321323 general problem, 331333 which changes a curve on the surface into a given curve in space, 333336 which preserves asymptotic lines, 336, 342, 343 which preserves lines of curva which preserves ture, 336338, 341 conjugate systems, 338342, 349, 350, 443 of ruled surfaces, 343348, 350,
surfaces)
(see
;
;
Element, of are.a, 75, 145 linear (see Linear element) normal sec Ellipsoid, equations, 228 tion, 234 polar geodesic system, 236238; umbilical geodesies, 236, 267; surface corresponding with par allelism of tangent plane, 269. See Quadrics
;
;
;
;
;
;
;
Elliptic coordinates, 227 Elliptic point of a surface, 125, 200 Elliptic type, of pseudospherical sur of surfaces of Bianchi, faces, 274 370, 371 Enneper (torsion of asymptotic lines), 140 (equations of a minimal surface)
; ;
,
;
256
Enneper, minimal surface
;
367; method of Weingarten, 353369 of the of paraboloids, 348, 368, 369 envelope of the planes of a cyclic
;
;
system, 429, 430
Developable
surface, definition, 61 ; particular kinds, 69 equation, 64 polar, 64, rectifying, 62, 64, 112, 209 applicable to the plane, 65, 112, 209 101, 156, 219, 321, 322 formed by nor mals to a surface at points of a line of curvature, 122 principal radii, 149
;
;
;
;
of, 269; sur faces of constant curvature of 317, 320 Envelope, definition, 59, 60 of a oneparameter family of planes, 6163, 64, 69, 442 of a oneparameter fam ily of spheres, 6669 of a twoparam eter family of planes, 162, 224, 426, 439; of geodesies, 221; of a twoparameter family of spheres, 391, 444 * parametric, 53;
, ;
;
;
;
Equations, of a curve, 52, 53, 54
1,
2,
52,
1, 2, 3,
21; of a surface,
;
;
250 geodesies on a, 224, 268, 318, 322 fundamental property, 244; of a congruence (see Congruence) Dextrorsum, 19 Differential parameters, of the first order,
total curvature, 156,
; ;
8488, 90, 91, 120, 160, 166, 186 of the second order, 8891, 160, 165, 166, 186 Diui (spherical representation of asymp totic lines), 192; (surf aces of Liouville), 214 (ruled TFsurfaces), 299 Dini, surface of, 291, 318 Directorcone of a ruled surface, 141 Directordevelopable of a surface of
;
Equidistance, curves of, 456 Equidistantial system, 187, 203 Equivalent representation of two sur faces, 113, 188 Euler, equation of, 124, 221 Evolute, of a curve, 43, 4547 of a surface, 180, 415 (see Surface of center) of the quadrics, 234 mean, of a surface, 165, 166, 372
; ; ;
F.
&
SeeE
See
.
;
//
Seee
Monge, 305
Directrix of a ruled surface, 241 Dobriner (surfaces with spherical lines of curvature), 315 Dupin (triply orthogonal systems), 449 Dupin, indicatrix of, 124126, 129, 150 cyclide of (see Cyclide) theorem of Malus and, 403
; ;
jE,
&&gt;
F, G, definition, 70 for the moving trihedral, 174 definition, 141 for the moving trihedral, 174 e,/,/, flr, definition, 393
;
&&gt;
^
;
Edge
of regression, 43, 60, 69
*
Family, oneparameter, of surfaces, 59, 446, 447, 451, 452, 457461; of planes, 6164, 69, 442, 463 of spheres, 6669, 309, 319, 463 of curves, 7880 of geo desies, 216, 221 Family, twoparameter, of planes, 162, 224, 426, 439 of spheres, 391, 444 Family of Lame", 461, 463, 464 Focal conic, 226, 234, 313, 314 Focal planes, 400, 401, 409, 416 Focal points, 398, 399, 425 Focal surface, of a congruence, 400 reduces to a curve, 406, 412 funda mental quantities, 409411 develop able, 412; met by developables in lines of curvature, 414 of a pseudospherical congruence, 416; infinitesi mal deformation of, 420 intersect, 423
; ;
;
;
;
;
;
;
;
For references such as Equations of Codazzi, see Codazzi.
4TO
Form
INDEX
Helicoid, general, 146148 parameter meridian of, 140 geodesies, of, 140 surfaces of center of, 151, 209 149, 186 pseudospherical, 291 is a IPsur
; ; ;
of a curve, 18 FrenetSerret formulas, 17 Fundamental equations of a congruence, 406, 407 Fundamental quadratic form, of a sur of a surface, second, face, first, 7 1 115; of a congruence, 393 Fundamental quantities, of the first order, 71; of the second order, 115 Fundamental theorem, of the theory of curves, 24 of the theory of surfaces, 159
;
;
;
;
300 minimal, 329, 331 appli cable to a hyperboloid, 347 Helicoid, right, 146, 148, 203, 247, 250, 260, 267, 330, 347, 381, 422 Helix, circular, 2, 41, 45, 203 cylindri
face,
; ;
;
;
G.
&
g.
See 8ee See e
60
;
E
cal, 20, 21, 29, 30, 47, 64 Henneberg, surface of, 267 Hyperbolic point, 125, 200 Hyperbolic type, of pseudospherical sur
Gauss (parametric form of equations),
representation), 141 (total curvature of a surface), 155 (geodesic parallels), 200; (geodesic cir cles) 207 (area of geodesic triangle)
(spherical
;
;
;
face, 273 of surface of Bianchi,371, 379 Hyperboloid, equations, 228 fundamen tal quantities, 228230; evolute of, 234 of revolution, 247, 348 lines of deformation of, 347, striction, 268 348. See Quadrics
; ;
;
;
;
209 Gauss, equations of, 154, 155, 187 Generators, of a developable surface, 41 of a surface of translation, 198 of a ruled surface, 241 Geodesic circles, 207 Geodesic contingence, angle of, 212 Geodesic curvature, 132, 134, 135, 13(5, 140, 213, 223 radius of, 132, 150, 151, center of, 132, 225, 174, 170, 209, 411 curves of 423 invariance of, 135 constant, 137, 140, 187, 223, 319 Geodesic ellipses and hyperbolas, 213215, 225 Geodesic parallels, 207 Geodesic parameters, 207 Geodesic polar coordinates, 207209, 230, 276 Geodesic representation, 225, 317 Geodesic torsion, 137140, 174, 176 radius of, 138, 174, 176 Geodesic triangle, 209, 210
;
Indicatrix, of Dupin (seel)upin); spheri cal (.see Spherical) Infinitesimal deformation of a surface,
;
373, 385387 generatrices, 373, 420 of a right helicoid, 381 of ruled sur in which lines of curva faces, 381 ture are preserved, 387, 391; of the focal surfaces of a TFcongruence, 420 Intrinsic equations of a curve, 23, 29,
; ; ;
;
;
30, 30
;
;
;
Invariants, differential, 8590 of a dif ferential equation, 380, 385, 406 Inversion, definition, 190 preserves lines of curvature, 190 preserves an isotherm ic system of lines of curva ture, 391 preserves a triply orthog onal system, 403. See Transformation
;
;
;
;
;
by reciprocal radii Involute, of a curve, 4345, 311 of a surface, 180, 184, 300 Isometric parameters. See Isothermic
;
parameters
,
*
Geodesies, definition, 133 plane, 140 equations of, 204, 205, 215219; on surfaces of negative curvature, 211 on surfaces of Liouville, 218, 219 Goursat, surfaces of, 306, 372
; ;
;
Isometric representation, 100, 113 Isothermalconjugate systems of curves, 198200; spherical representation, 202 formed of lines of curvature, 147, 203, on associate surfaces, 300 233, 278 Isothermal orthogonal system. See Iso
;
;
Guichard (spherical representation of
the developables of a congruence), 409; (congruences of Ribaucour), 421 Guichard, congruences of, 414, 415, 417, 422, 442
I/, definition,
//, definition,
71 142
of, 278, 279,
thermic orthogonal system Isothermic orthogonal systems, 9398, 209, 252, 254 formed of lines of curva ture (see Isothermic surface) Isothermic parameters, 9397, 102 Isothermic surface, 108, 159, 232, 253, 269, 297, 387389, 391, 425, 465
;
Isotropic congruence, 412, 413, 416, 422
Hamilton, equation of, 397 Hazzidakis, transformation 338
424
Isotropic developable, 72, 171, 412, 424 Isotropic plane, 49
p. 467.
*
See footnote,
INDEX
Jacob! (geodesic lines), 217 Joachimsthal (geodesies and lines of curvature on central quadrics), 240 Joachimsthal, theorem of, 140 surfaces of, 308, 309, 319
;
471
Meridian curve on a surface, 260 Meusnier, theorem of, 118 Middle envelope of a congruence, 413, 415 Middle point of a line of a congruence,
Kummer
Lame
(rectilinear congruences), 392
Lagrange (minimal surfaces), 251
85 equations of, 449 family of, 401, 463, 464 Lelieuvre, formulas of, 193, 195, 417, 419, 420, 422 Lie (surfaces of translation), 197, 198 (double minimal surfaces), 259 (lines of curvature of JFsurfaces), 293 Lie, transformation of, 289, 297 Limit point, 396, 399 Limit surface, 389 Line, singular, 71 *Line of curvature, definition, 121, 122, 128; equation of, 121, 171, 247; par normal cur ametric, 122, 151, 186 vature of, 121, 131 geodesic torsion of, 139 geodesic, 140 two surfaces inter secting in, 140 spherical representa
(differential parameters),
;
399 Middle surface of a congruence, 399, 401, 408, 413, 421424 Minding (geodesic curvature), 222, 223 Minding, problem of, 321, 323, 326
;
Lame",
method of, 344 Minimal curves, 6,
;
47, 49, 255, 257 on a surface, 81, 82, 85, 91, 254265, 318, 391 on a sphere, 81, 257, 364366
;
;
390
;
Minimal straight lines, 48, 49, 260 Minimal surface, definition, 129, 251;
asymptotic lines, 129, 186, 195, 254, 257, 269 spherical representation, 143, 251254; ruled, 148; helicoidal, of revolution, 160 149, 330, 331 parallel plane sections of, 160 mini mal lines, 177, 186, 254265; lines of curvature, 186, 253, 257, 264, 269 double, 258260 algebraic, 260262 evolute, 260, 372 adjoint, 254, 263,
; ; ;
;
;
;
;
;
;
;
;
;
;
tion of, 143, 148, 150; osculating plane, 148; plane, 149, 150, 201, 305314, 319, 320, 463 plane in both systems, 269, 300304, 319, 320 spherical, 149, 314317, 319, 320, 465 circular, 149, 310314, 316, 446; on an isothermic surface, 389, Line of striction, 243, 244, 248, 268, 348, 351, 352, 369, 401, 422 Linear element, of a curve, 4, 5 of a surface, 42, 71, 171; of the spherical
;
; ; ;
267, 377; associate, 263, 267, 269, 330, 381; of Scherk, 260; of Henneberg, 267; of Enneper, 269; deformation
of, 264,
327329, 349, 381
;
determi
;
geodesies, 267 Molding surface, definition, 302 equa tions of, 307, 308 lines of curvature,
of, 265,
; ;
nation
266
307, 308, 320 applicable, 319, 338 associate to right helicoid, 381 nor mal to a congruence of Ribaucour,
;
;
;
422
Molding surfaces, a family
465
of
Lame"
of,
representation, 141, 173, 393; reduced form, 353 of space, 447 Lines of length zero. See Minimal lines Lines of shortest length, 212, 220 Liouville (form of Gauss equation), 187 (angle of geodesic contingence), 212 Liouville, surfaces of, 214, 215, 218, 232
;
;
(equations of a surface), 64 (molding surfaces), 302 Monge, surfaces of, 305308, 319 Moving trihedral for a curve, 3033 applications of, 3336, 39, 40, 6468 Moving trihedral for a surface, 166170
Monge
;
;
;
Loxodromic curve,
140, 209
78, 108, 112, 120, 131,
rotationsof, 169; applications of, 171183, 281288, 336338, 352364, 426442
Mainardi, equations
of,
156
of
Normal, principal,
allel to
definition, 12;
16,
par
Malus and Dupin, theorem of, 403 v. Mangoldt (geodesies on surfaces
a plane,
;
21
Normal congruence
gruence)
of lines (see
Con
Mean Mean Mean
positive curvature), 212 curvature, 123, 126, 145 evolute, 165, 166, 372 ruled surfaces of a congruence,
Normal curvature
Curvature
of curves (see Congruence) of a surface. See
422, 423, 425 Mercator chart, 109
Meridian, of a surface of revolution, 107; of a helicoid, 146
*
Normal plane to a curve, 8, 15, 65 Normal section of a surface, 118, 234 Normal to a curve, 12 Normal to a surface, 57, 114, 117, 120,
121, 141, 195
p. 467.
See footnote,
472
INDEX
;
Normals, principal, which are principal normals of another curve, 41 which are binormals of another curve, 51
Order of contact, 8, 21 Orthogonal system of curves,
;
75,
77,
8082, 91, 119, 129, 177, 187; par ametric, 75, 93, 122, 134 geodesies, 187 isothermic (see Iso thermic) Orthogonal trajectories, of a oneparam of a eter family of planes, 35, 451 family of curves, 50, 79, 95, 112, 147, of a family of geodesies, 149, 150 216 of a family of surfaces, 446, 451, 452, 456, 457, 460, 463, 464
;
;
Point of a surface, singular, 71 elliptic, 125, 200 hyperbolic, 125, 200 para middle bolic, 125; focal (see Focal) (see Middle); limit (see Limit) Polar developable, 64, 65, 112, 209 Polar line of a curve, 15, 38, 46 Principal directions at a point, 121 Principal normal to a curve. See Normal Principal planes of a congruence, 396, 397
; ; ; ;
Principal radii of normal curvature,
120, 291, 450
lit),
;
Principal surfaces of a congruence, 390398, 408
Projective transformation, preserves os culating planes, 49 preserves asymp totic lines and conjugate systems, 202 Pseudosphere, 274, 290 Pseudospherical congruence, 415, 416, 464 normal, 184
;
;
Osculating circle, 14, 21, 65 Osculating plane, definition, 10
; ;
;
;
equa
tion of, 11 stationary, 18 meets the curve, 19 passes through a fixed point, 22 orthogonal trajectories of, 35 of edge of regression, 57 of an asymp of a geodesic, 133 totic line, 128 Osculating planes of two curves parallel,
;
; ; ;
;
50 Osculating sphere, 37, 38, 47, 51, 65
Parabolic point on a surface, 125 Parabolic type, of pseudospherical sur of surfaces of Bianchi, faces, 274
;
Pseudospherical surface, definition, 270 asymptotic lines, 190, 290, 414 lines of curvature, 190, 203, 280, 320 geo defor desies, 275277, 283, 317, 318 mation, 277, 323 transformations of, 280290, 318, 320, 370, 45(5 of Dini, 291, 318; of Enneper, 317, 820; evo lute, 318 involute, 318 surfaces with
; ; ;
;
;
;
;
;
370, 371, 442, 443, 445 Paraboloid, a right conoid, 56 tangent plane, 112 asymptotic lines, 191, 233; a surface of translation, 203 equa fundamental quanti tions, 230, 330 lines of curvature, 232, 240 ties, 231 evolute of, 234 of normals to a ruled line of striction, 268 surface, 247
; ;
the same spherical representation of their lines of curvature as, 320, 371, 437, 439, 443, 444. See Surface of
;
;
;
;
;
constant total curvature Pseudospherical surface of revolution, of elliptic of hyperbolic type, 273 of parabolic type, 274 type, 274 Pseudospherical surfaces, a family of Lam6 of, 452456, 464
; ;
;
;
deformation of, 348, 349, 367369, 372 congruence of tangents, 401. See Quadrics on a surface Parallel, geodesic, 86, 207
; ;
of revolution, 107
Parallel curves, 44 lines Parallel surface, definition, 177 of curvature, 178 fundamental quan cur tities, 178; of surface of constant vature, 179 of surface of revolution,
;
Quadratic form. See Fundamental Quadrics, confocal, 226,401 fundamen tal quantities, 229 lines of curvature, asymptotic lines, 233 233, 239, 240 geodesies, 234236, 239, 240 associate normals to, 422. surfaces, 390, 391 See Ellipsoid, Hyperboloid, Paraboloid
;
;
;
;
;
;
;
;
185
Parallel surfaces, a family of
Lame"
of,
446 Parameter, definition,
1 of distribution, 245, 247, 268, 348, 424, 425 Parametric curves, 54, 55 Parametric equations. See Equations Plane curve, condition for, 2, 16 curv ature, 15 equations, 28, 49 intrinsic
;
;
;
;
equations, 36
Plane curves forming a conjugate sys tem,. 224 Plane lines of curvature. See Lines of
curvature
Representation, conformal (see Conformal); isometric, 100, 113; equiv alent, 113, 188; Gaussian, 141; conformalcon jugate, 224 geodesic, 225, 317 spherical (see Spherical) Revolution, surfaces of. See Surface Ribaucour (asymptotic lines on surfaces of center), 184 (cyclic systems of equal circles), 280; (limit surfaces), 389 (middle envelope of an isotropic (cyclic systems), congruence), 413 (deformation of the 426, 428, 432 of the planes of a cyclic envelope (cyclic systems system), 429, 430 associated with a triply orthogonal system), 457
; ;
;
;
;
;
;
INDEX
Ribaucour, congruence of, 420422, 424, 425, 435, 442, 443 triple systems of, 452, 455, 463 Riccati equation, 25, 26, 50, 248, 429 Rodrigues, equations of, 122 * Ruled surface, definition, 241 of tan gents to a surface, 188 generators,
; ;
;
473
Spherical representation of an axis of a moving trihedral, 354
241; directrix, 241; linear element, line of 241, 247 directorcone, 241 striction, 243, 244, 248, 268, 348, 351, 352, 309, 401, 422 central point, 243 central plane, 244 parameter of dis tribution, 245, 247, 208, 348, 424, 425 doubly, 234; normals to, 195, 247; total tangent plane, 246, 247, 268
; ;
;
;
;
Spherical surface, definition, 270; par allels to, 179 of revolution, 270272 geodesies, 275279, 318 deformation, 323 lines of curvature, 278 276, invo transformation, 278280, 297 of Enneper, 317 surface lute, 300 with the same spherical representation of its lines of curvature as, 338. See Surface of constant total curvature Spherical surfaces, a family of Lame" of,
; ; ; ; ;
;
;
;
;
463
Spiral surface, definition, 151 gener lines of curvature, 151 ation, 151 minimal lines, 151 asymptotic lines, 151 geodesies, 219 deformation, 349
;
;
;
;
curvature, 247 asymptotic lines, 248250 mean curvature, 249 lines of curvature, 250, 268 conjugate, 268 deformation, 343348, 350, 367; spher ical indicatrix of, 351; infinitesimal of a congruence, deformation, 381 393395, 398, 401, 422, 423. See Right conoid, Hyperboloid, Paraboloid
;
;
;
;
;
;
;
;
;
Scheffers (equations of a curve), 28 Scherk, surface of, 260
Schwarz, formulas of, 264267, 269 Singular line of a surface, 71 Singular point of a surface, 71
Sinistrorsum, 19 minimal Sphere, equations, 62, 77, 81 conformal representation, lines, 81
; ;
109111 equivalent representation, 113 fundamental quantities, 116, 171; principal radii, 120; asymptotic lines, 223, 422 Spheres, family of. See Family Spherical curve, 36, 38, 47, 50, 149, 314316, 317, 319, 320, 465 Spherical indicatrix, of the tangents to a curve, 9, 13, 50, 177 of the binormals to a curve, 50, 177 of a ruled surface, 351 Spherical representation of a congruence, definition, 393 principal surfaces, 397, 408 developables, 409, 412414, 422, 432435, 437, 441 Spherical representation of a surface, fundamental quan definition, 141 lines tities, 141143, 160165, 173;
; ;
;
Stereographic projection, 110, 112 Superosculating circle, 21 Superosculating lines on a surface. 187 t Surface, definition, 53 Surface, limit, 389 met \ Surface of center, definition, 179 by developables in a conjugate sys tem, 180, 181 fundamental quantities, 181, 182 total curvature, 183 asymp totic lines, 183, 184 lines of curva a curve, 186, 188, 308ture, 183, 184 314 developable, 186, 305308 Surface of constant mean curvature, definition, 179 parallels to, 179 lines of curvature, 296298 transforma tion, 297 deformation, 298 minimal curves, 318 Surface of constant total curvature, area of geodesic tri definition, 179 lines of angle, 219 geodesies, 224
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
;
curvature, 317; asymptotic lines, 317; See spherical representation, 372. Pseudospherical surface and Spheri cal surface
;
;
;
of curvature, 143, 148, 150, 151, 188, 201, 253, 279, 280, 292, 296, 301, 302, 308, 314, 315, 320, 371, 387, 437, 442445 asymptotic lines, 144, 148, 191195, 254, 340, 390, 414 area of closed portion, 145 conjugate system, 200202, 257, 385
; ;
;
Surface of reference, 392 Surface of revolution, definition, 107 fundamental quantities, 107, 147 loxodromic curve (see Loxodromic) deformation, 108, 112, 147, 149, 260, 276, 277, 283, 326331, 341, 349350, 362364, 369, 370, 372, 444; partic ular, 111, 160, 320 equivalent repre sentation, 113 lines of curvature, 126 parallel sur asymptotic lines, 131 faces, 185 geodesies, 20.5, 209, 224 Surface of translation, definition, 197, 198; equations, 197; asymptotic lines, 198; generators, 198, 203; deformation, 349, 350 associate surface, 381, 390;
;
; ; ; ;
;
;
;
;
reference is to nondevelopable ruled surfaces. For developable ruled surfaces, see Developables. t For references such as Surface of Bianchi, see Bianchi. I Surfaces of center of certain surfaces are referred to under these surfaces.
*
Tim
474
congruence of tangents, 406
surface of a
&gt;Fcongruence,
;
INDEX
middle
422, 424
Surface with plane lines of curvature. See Lines of curvature Surface with spherical lines of curvature. See Lines of curvature Surface with the same spherical repre sentation of its lines of curvature as a pseudospherical surface. See Pseudospherical surface Surface with the same spherical repre sentation of its lines of curvature as a spherical surface. See Spherical sur
face
* Transformation, of curvilinear coordi of rectangular nates, 5355, 73, 74
;
coordinates, 72
;
by reciprocal
;
radii,
104, 196, 203 (see Inversion) ive (see Projective)
project
Surfaces of revolution, a family of of, 451
Lame"
Triply orthogonal system of surfaces, definition, 447; associated with a cyc lic system, 440 fundamental quan with one family of tities, 447451 surfaces of revolution, 451, 452 of Kibaucour, 452, 403 of Bianchi, 452of Weingarten, 455, 454, 404, 465 transformation of, 462, 456, 403, 404 463 with one family of molding sur faces, 405 of cyclides of Dupin, 405 of isothermic surfaces, 405
;
; ;
;
;
;
;
;
;
Tangent plane
to a surface, definition, 50, 114; equation, 57; developable sur meets the face, 67; distance to, 114
; ;
;
Umbilical point of a surface, definition, 120 of quadrics, 230, 232, 234, 230238, 240, 207
;
characteristic of, 126 surface, 123 is the osculating plane of asymptotic
128 Tangent surface of a curve, 4144, 57; applicable to the plane, 101, 150 Tangent to a curve, 6, 7, 41), 50, 51) spherical indicatrix of, 9, 13, 50, 177 Tangent to a surface, 112 Tangential coordinates, 103, 104, 201 Tetrahedral surface, definition, 207 asymptotic lines, 207 deformation, 341 Tetrahedral surfaces, triple system of, 465 Tore, 124 Torsion, geodesic, 137140, 174, 170 Torsion of a curve, definition, 10 radius of, 16, 17, 21; of a plane curve, 10 sign of, 19; constant, 60; of asymp totic line, 140 Tractrix, equations, 35 surface of revo
line,
;
Variation of a function, 82, 83 Voss, surface of, 341, 390, 415, 442, 443
Wcongruence, 417420, 422, 424 fundamental &gt;Fsurface, definition, 291 quantities, 291293 particular, 291,
;
;
;
300, 318, 319; spherical representation, 292; lines of curvature, 293; evolute, 294, 295, 318, 319; of Weingarten, 298, 424; ruled, 299, 319 Weierstrass (equations of a minimal sur face), 200 (algebraic minimal sur faces), 201
;
;
Weingarten
103
las),
;
(tangential (geodesic ellipses
coordinates).
and hyperbo
;
;
;
214; (&gt;Fsurfaces), 291, 292, 294 (infinitesimal deformation), 374, 387 (lines of curvature on an isothermic
; ;
;
lution of, 274, 290
;
helicoid
whose
meridian
is a,
291
as
surface), 389 Weingarten, surface of, 298, 424 method of, 353372 triple system of, 455, 450, 403, 464
;
*
For references such
Transformation of BJickluud, see Bitcklund.
"
A*tr*

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r
UBfURY