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TREATISE mmm Wf / I 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 89-8 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 Frenet-Serret 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 II-YI, 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 subject-matter 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. FRENET-SERRET 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 ONE-PARAMETER 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 ISOTHERMAL-CONJUGATE 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. TF-SURFACES. 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 JF-SURFACES. 289 291 FUNDAMENTAL QUANTITIES . 124. 125. 126. 127. 128. 129. EVOLUTE OF A TF-SuitFACE SURFACES OF CONSTANT MEAN CURVATURE RULED IF-SuRFACES 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 IT-CONGRUENCES 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>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 direction-cosines 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>axis, 2-axis, 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 </>, 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?y-plane, 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) 2-axis and whose cross section by the xy-pl&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, <S> 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<( w*+i ~ %) differentiation. < ^ < * 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 direction-cosines 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 . /*->. = = > /!*) 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 direction-cosines. The (23) distance from {[(bx>- M l to this line is equal to \(bx"- ay")e* ay )e + + 1 2 ] 2 . bz )e + 2 -] + [(az - cx )e + -] }*. Hence, if MM l be considered an infinitesimal of the first order, this distance also is of the first order unless in which case it is of the second order at least. But when these equations are satisfied, equations (22) define the tangent at M. Therefore, of all the lines through a point of a curve the tangent is nearest to the curve. , , * Whenever the functions x y z appear in a formula it is understood that the arc s is the parameter otherwise we use /{, /2 /3 indicating by accents derivatives with respect to the argument u. ; , , 8 5. CURVES IN SPACE Order of contact. Normal plane. When the curve is such that there are points for which (24) ^=4 x y z the distance from to the tangent is of the third order at least. l In this case the tangent is said to have contact of the second order, whereas, ordinarily, the contact is of the first order. And, in gen eral, the tangent to a curve has contact of the wth order at a point, M if the following conditions are satisfied for n = 2, , n 1, and n : < 25 ) ^=^=^rr Jl fl ~jTf xw i(0 jyV V 5>V"y the parameter of the curve is any whatever, equations (24), (25) are reducible to the respective equations f (rt-1) 2, When J% J% // J\ -f(.n-\) J Ja f(n-l) 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 cross-ratio. 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 one-to-one 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<9 = 1. da Ao- Hence we have (27) = ds , where da- is The the linear element of the spherical indicatrix. coordinates of are the direction-cosines 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 W-fifl+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 + (Z-z)c = Q, the current coordinates. When the Jf, through the tangent at (33) If the values (17) for the coefficients a, c >, 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 >, from equations (32), (33), (34) we obtain, as the equation of the osculating plane, X-x Y-y Z-z (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, X-x Yx Z-z = 0. y plane, and its plane is taken for the rry-plane, 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 direction-cosines of the binormal be denoted by X, />t, z>, 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-, z-axes 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 tion-cosines of the principal normal, we have* (39) m /JL =4-1, 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 direction-cosines 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 1-r^Z/ where 77 +*? = (), involves terms of the limi-t 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\ X-x-pl = Y-y-pm = 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 cross-ratio is constant. and four fixed points of the curve (c) Four planes through a variable tangent Four are in constant cross-ratio. (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 <f> so that the principal normals to the (u) are parallel to the yz-plane. 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. Frenet-Serret 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 direction-cosines a, & m, w; X, p, v. We deduce them now. (4T) From a (41) /3 we have =i, =, y-. FRENET-SERRET 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 ,: > = T t V = T s l*> 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 Frenet-Serret 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 direction-cosines 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-, 2-axes 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 \<JO < 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 direction-cosines 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 z-coordinate 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 near-by 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 3n-cosines 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 </> d<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 Frenet-Serret formulas for the two curves are ds ds r ds -=ds = = p > = ds ds ( -| -- I = r -> \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 >) + (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 , <*,-,) = 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 />=/,<), 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 direction-cosines 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 u-r+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 direction-cosines 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 direction-cosines 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\* /^>\ 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 direction-cosines of the principal normal in the positive sense. In consequence of (40) the functions X, yu., v are the direction-cosines 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 o-ft) = ,1 i + o-ft) 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. 173-177. 26 CURVES IN SPACE we put Let O l be a particular integral of (66). If is the equation for the determination of </> = l/c -f 0^ - (67) + 2(M+2WJ<l>+N**Q. and of the first As this equation is linear order, it can be solved by two quadratures. <=/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 !/< ty/<l>)=N(0 1 2 )^/<t>. Consequently the general integral of cts (66) is given by 69 < > 00, -v|r 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 cross-ratio as these constants. Hence we have the theorem The cross-ratio 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> (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 , &> ; <7 , &> 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 &) <T., . t are substituted in the expressions (73) 7 = ,1+a, a where, for the sake of brevity, we have put (74) RS PQ PS-QR 28 CURVES IN SPACE The coefficients of U, F, and W in , (73) are of the z>. 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 direction-cosines 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 w-K ) ci I/ RS-/~r> 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~ <r i<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<rds, y=Csir\<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_2c0-02). 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 <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 * = -|=-* <x> 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 z-axis is And the cross-section 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 cross-section 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 z-axis 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 direction-cosines of PP^ with respect to the axes at Jf, then If a, 5, <? 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 left-hand members of equations (82) and the equations reduce to (84) ds --l, p = ds \p T/ c?s T Moreover, the direction-cosines 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. 122-128. MOVING TRIHEDRAL These are the Frenet-Serret 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 direction-cosines, 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*)> 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 direction-cosines 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 direction-cosines of the principal normal of C\ are 5/?i 571 By means of (40) : we derive the following expressions for the direction-cosines of the binormal r (a 2 + p )^ 2 r (a 2 + p )^ 2 ft2 > + ? 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. <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 mid-point 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, > 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 < it follows that when r > > 0, co is in < < the third, < 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 left-hand 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 direction-cosines 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 direction-cosines of the same direction, with respect to the &> 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, <? a>, 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) <w to sin PI cos TI &)1 I ~ [T */> TI J L ~| , * , u, J dco ~ &) /3 [sin cos &) k . 6 -h , (<? 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 ;; <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> where B, C are constants different from zero ; if we take k =A , COt ft) = B ;> 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<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 < 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), #! =#+ (<?), zx =z+ ; ?(<? 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> Eliminating /c, we get p For the sake of convenience we put integration <o = I - > and obtain by P = tan (o> + 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> -f c), Q -f- (o> 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 = &> + 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.<l+*>:.. 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, 1-a 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 direction-cosines are proportional to > 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 >()<**, 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) 1y-4 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, + (Y-y)B + (Z-z)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 x-axis, 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 x-axis. 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 + + 1-2 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> 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 one-to-one 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 one-to-one 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<f> u du, y = ^ 2 a ) \ $ (u) cos u du, and \f/ z a f 4>(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 <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 vida-i \ (v\\{ v{\\] d<?i, ek | (\i/4 where k and w are constant, e = 1, and the primes indicate differentiation with respect to o-i, define a Bertrand curve for which p and T satisfy the relation (97) show also that X 1? /t 1? v\ are the direction-cosines 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 z-axis FIG. 10 of the sphere makes and a point with the #z-plane, and u denotes the angle between the radius OM and the positive z-axis, 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 z-axis, 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 = <t>(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> 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> 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^ = <$>! (w, v), v^ = 4> 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 = <j> (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>, y Ma 4- sin v, z = w, determine the that two and second curves whose tangents make with the z-axis 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) f-s = 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 = * <f>(u), the above equations may be written . ( ^ , , \ dx\ j du dvl ds =^ \ \du -f- ^^ 4> 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-^+fo-jO^+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 -(f-z) = 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 g-X^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 near-by is of the first order and for other planes MM ; ordinarily of the first order. 26. One-parameter 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 one-parameter 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 > * 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 F-0, ^=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 direction-cosines 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 right-hand mem ber are proportional to the direction-cosines 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 X-x _Y-y = 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 one-parameter 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 (jL--J\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 C-X^ 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 one-parameter family of planes parallel line. 5. to a given 6. stant angle Given a one-parameter family of planes which cut the xy-plane 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 < > !( - 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 > rVl 1. r n . Hence the char 1, reduces to a point when + const., and is real for r f * < 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 <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^l-r \ 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 right-hand 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 direction-cosines of the tangent to C^ we see that this tangent is normal to the osculating plane to the curve of centers C. Moreover, these direction-cosines 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 direction-cosines 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 direction-cosines 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 = : > ? + 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>, 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 z-axis. 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, <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 <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 <^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 right-hand 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>(u) du + u<j> (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 < 1 __ F (faito1 cu^ ct\ i- du cu cu EG- r^ ,T /" 1 cv JG/Cr Tfi S~1 CM Jj<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 direction-cosines of the we have fa du 1 cu cy "*</ E du dx < 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 &) = VJSQ--F* 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>. \ 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 direction-cosines 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< _ J If ~ dz _ du cz dv ~ds ds~\^uds dv we put dv/du = X and the right-hand 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 direction-cosines 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 direction-cosines 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<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>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) <(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 i|r 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<f> g*H.2*.i.<* 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 <. a function of </>. Moreover, if ^r can But by any function of of the families of curves <, we have seen that -fy = const, </> and = const, ^ is a function are the same. Hence admit all integrals of equation (30) of the form <f>=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>, 0. But if <?, 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 > 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 x-axis 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 y-axis. 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(l-v 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 X-x, , Y-y. Z-z, point. where # y^ z are the coordinates of a particular manner the curves u = const, are the minimal lines In like X-x. Y-y. = Z-z, , 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. <j> When the values of the coordinates of a point <, 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> 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 right-hand 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 <f> 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 <f>= from the point. It equals c, and attains its greatest value being normal to this curve, this 4 < > ; S m^\-ZF-z-z-+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 <= 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> 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 mid-point of the = segment, measured along the tangent to v t const., The maximum is found to be - readily =<> 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 <> -** differential quotient corresponds to the direction The left-hand 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 right-hand 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 < : 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 EG-F 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^ = </> and a solution = const. From the latter we get, by differentiation, d<l> , - 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 (/> </> a solution of equation (42). Another particular case is that in which A^ is a function of <, say (43) * A,* les = *<*). 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 </> 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 = (/> the curves of a family necessary and sufficient condition that be a function of const, be parallel is that \(f> <f>. 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> and = const. Now equation (22) can be written r \du dv dv c which by means of the function <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 (<^, known It is an invariant. a mixed (49) it differential parameter of the 2 ((/>, first order. From (47) and (50) follows that 2 A, (<, 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< AlV = ^ jfi (51) (u, v) = A,H V-A 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>(Mdv-Ndu)- ff$(^+ j^} dudv > *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 ) > 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/> 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>, AA(4>.M>)> 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/>, , Conversely, T ftw /== /(A we shall show -, * that every invariant of the form v ^ . . r dE tr, dG -, di> *W dw dw 0, -^-, I -, i/r, -^L, ^ du ), where of the . <, -^, 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 </> ty respectively, change and con to these sequently we express / in terms of (/>, ^, and the differential <*) invariants obtained by applying the operators A and functions. In case $ is the only function appearing in /, c/>, we can such as take for i/r, 2 <, in the change of parameters, any invariant of it is A^ or A so long as not a function of (/>, E, F> 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 right-hand 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 x-axis of any displacement on the surface is given by dx dx =- d\b = + dx * dd> 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>, 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, ()=<), = 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 right-hand 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 > da> t Eliminating t from these equations, we have E^-F^ du dv 63 < > -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 = <+ty, = </> iyfr. If these values be substituted in (59), we get (67) ds* = \(d<t> 2 At once we see that the curves +d^). = const, c/> and ^r = const, form an orthogonal system. lines are V\d-*fr Moreover, the elements of arc of these increments (<f) d<f> and respectively. Consequently when the and d^ are taken equal, the four points i/r), ^\d(f> (<, -f c?<, i/r), ($, i/r -f efo/r), -f (< tity, = const, and small square. Hence the curves the surface into a network of small squares. (f> ^ -f eityr) are the vertices of a ^|r = const, this divide On and account these curves are called isometric curves, and <f> 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 : <, Given any pair of real isometric parameters every other pair < -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 Cauchy-Riemann differential equations ?*i d<l> = *i, c^ ?& = _?*i, 3^ 8<f> < it follows that </> (f) l and the curves <f> 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 right-hand 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 <f> 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 <f> (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 </>+ i^r, and f the func A *=0, when we have a function (f> 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 (<o) H = dd> dv "> du du H ( ) = d<f> 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> express the condition = > ( ) we find that parameters. Hence : A the necessary and sufficient condition that be the isometric <f> param c/> 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 <> i-f 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> 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> now function (w, v) must the necessary and sufficient condition which a = const. satisfy in order that the curves o> their o>, orthogonal trajectories it, or a function of is form an isothermic system. the isothermic parameter of the then </>; o> = const. We A o> denote this parameter by . </>=/() Since </> must be a solution 2 of equations (71), we have, on substitution, (G>) (76) ./ + \a> ./"(a>) = 0, to &>. 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>, is a func tion of /(a>), co. Conversely, if this ratio is a function of obtained by two quadratures from the function (77) / necessary and const, is (*>) = *-/>, condition that will satisfy equations (71). Hence: a family of curves isothermic sys &>. A a) sufficient = and their orthogonal trajectories form an tem that the ratio of A &> &> 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> dv TT- du H * v r\ i , . wu -TT- t/i/ O <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 __#&> <f, ^cto s^fo -rJo<* 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> = 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 > 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 <f> = 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 one-to-one 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 <&? By r hypothesis a> =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 <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 = <-ty, a1 =<^ 1 +i> 1 , ft=0 -*^ 1 1, the functions fa and fa, ^ are > pairs of isothermic parameters. Now (88) equations (86), (87) <#> 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 = <j> 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> 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> For, the right-hand 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 =0-0 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 xy-plane, are represented parallels x circles which pass through the origin and have their centers the respective axes. Conversely, these circles in the xy-plane correspond to = z^-plane by the parallels in the Xi^/i-plane. If we put o;2 + y* = r2 , x* + y* = rf, equations (ii) and (iii) may be written <*> ?-? f-S. 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 vector-es. 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 x-axis 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) <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 " />" = 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 > 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 z-axis 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 <n-plane which corresponds to the circle of center (c, d) and radius r in the or-plane ? * 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 a-plane through and touching the other the origin corresponds to a circle in the a^plane through . circle ; also that a circle touching the x-axis 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 ai-plane. in the a-plane correspond to confocal ellipses = logo: to lines parallel to the x- and 7. In the conformal representation i y-axes in the ai-plane there correspond lines through the origin and circles about it in the a-plane, and to any orthogonal system of straight lines in the ai-plane an orthogonal system of logarithmic spirals in the a-plane. 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>axis and ?/-axis We take the axis of rotation for the 2-axis, and to the z-axis, 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 2-axis. = </>(r), denote the angle which the plane, in any of its positions, makes with the #2-plane. Hence the equations of the surface are let v We (99) x = r cos,v, is 2 y = rsinv, 2 z=(f>(r). The linear element (100) If ds = [1 + < (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 <f>(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 z-axis. 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/3-l 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 < + 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 zz-plane. = , 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> V az =<* 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 1-Z 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 z-axis 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> 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 <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 cross-ratio of four tangents to one surface at a point the corresponding tangents to the other. 2. a one-to-one point correspondence between two surfaces, the is equal to the cross-ratio 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 xy-plane. Show that the generators of the developable 2, enveloped by these planes, make a constant angle with the z-axis, 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 xy-plane 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 <f>(o) so that the right conoid, x = ucosv, y=usinv, z = (f>(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 ai-plane correspond to lines through a point (a, b) and circles concentric about it in the a-plane. 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 a-plane correspond to confocal ellipses and hyperbolas in the ai-plane. 10. In the conformal representation of the plane upon itself, given by ai = a2 , to lines parallel to the axes in the ori-plane there correspond equilateral hyperbolas in the a-plane, and to the pencil of rays through a point in the ori-plane and the cir about it there corresponds a system of equilateral hyperbolas through the corresponding point in the or-plane 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 <f> u (w, v), = $ (u, = M, v =v define an equivalent rep H \[<) 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 y-axis 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 direction-cosines. 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 2-axes. 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>, (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 <1> 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 left-hand 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 FD-ED H* FZ> 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> 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 right-hand 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> 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 right-hand 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 < o> < TT/%, and to pn 7r/Z<a><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 = > 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 right-hand Since the left-hand 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 + Ft-R(D + D = Q, = 0. \F+Gt-R(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" + GD-2 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 > , D" = where 2. p Show = dz > 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>, 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 z-axis. /> 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 > 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" ^ JL-i+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 > 0. Hence the distance from a near-by point to the tangent plane at Jf, since it is proportional to the fundamental form ( 48), does not <l> 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>du*+2 D dudv + is D" dv 2 = 0, which the normal curvature zero. In these directions the dis tances of the near-by 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 < 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. <1> 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 (> 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> 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 e-2_ 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 near-by 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 = -(tt-M), , 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 + p-2) 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 <(%, 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 one-parameter family of curves. These curves and the curves const, are said to form a conjugate system. Moreover, = </> 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 < ar tan 6 = x N^ Su [GSv - , so that equation (36) (39) may be put in the form tan0tan0 = -?H, is which 55. the well-known 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 self-conjugate. 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 self-conjugate, 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-/>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 right-hand 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 x-axis belongs to one of the latter families. their projections 9. Find the asymptotic lines on the surface 2 on the xy-plane. - 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 2-axes. From this choice of direction it follows that if, as usual, the direc tion-cosines 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~-*~> ds f ds *f-4ds 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 direction-cosines ( 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 <*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/> < 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 <= 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/ FM-EN 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> follows that o) = X\ -f Y/JL + Zv, If this where X, /Lt, v are the direction-cosines 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 . -V-K / . 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 hf-F 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 right-hand 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 _ FD-ED ~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 direction-cosines 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 direction-cosines 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> &>!, 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 = -(u-v), , / lines on the surface uv > 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 one-to-one correspondence. the sphere is called the spherical map upon representation of the surface, or the Gf-aussian 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 direction-cosines 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 > plied respectively by - ?., dv _ and dv du du -- and added, and du likewise by - > dv the an(l added, th resulting equations may be written D du +D dv - r(fdu+di>) = 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^ (<o 143 the equation of the Again, the elimination of du and dv gives 2 ) r*(<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 half-tangents 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 direction-cosines of the normal to the sphere are equal to X, Y, Z, to within sign at most. be denoted by then I/, ^; X><> Let them (79) x> = #\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> 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 2-axis, 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 #z-plane 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 = <f> of the surface are (81) x = u cos v, y = u sin v, z = </> (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 < (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<t> ^= 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. < perpendicular to the axis or oblique, according as is (/> 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 + ^ / >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<f> cos v, (a cos v 2 + u$ 2 ! sin v), u V^ 2 (l + f )+a />,/> .!>"= 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 </> + u<t> $ ] dii 2 - (1 + $ *) u^ + [(u + a +u +a ]dv =Q. -a[u 2 2 ) M<" dudv 2 - 2 2 2 2 (t> Moreover, the meridians are lines of curvature of those helicoids, for which </> satisfies the condition 148 GEOMETRY OF A SURFACE ABOUT A POINT integration this gives By = <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 = y-axis respectively. 6. x + pz=<t>(p) have a system of lines of curvature in planes parallel to the xz-plane 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 R-s, which make equal angles 2 tr/m with one another, and m 2, then a surface 9. If EI, , , > of I /-I m \Ri 10. If the V--1 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 > 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<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 z-axis, 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 z-axis 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 !/^ + ^_<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< I ~l I L * Crelle, Vol. t . J- I cv Q/* O L^J I I ^ 2~dv LXX, pp. 241-245. 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 4i--Jv du f) ^ =:^ du A =-!r* dv TT 2 fill I du ," dv Ti-O 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 > (MVKff}> From the above definition of the symbols : Sm-K?))f^l <! > 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 G-auss : 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 <* (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 , >, (10) !=0, a 2 =0, &j=0, 62 =0, ^=0, <? 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 HiAaJ-ta-J 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 G-auss 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 direction-cosines 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>, 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 right-hand 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 direction-cosines 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>), 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. 122-124. 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 direction-cosines of the normal to a surface, the direction-cosines 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 =-< ^ = -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 > 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> 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\ <o 01 dl ) -r rf \ , du C^\- dv / zr< ) rC rrV^ v I ~Y j^G-A- 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>, 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 >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. 234-248. I, pp. 172-174 . ; ; 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 /> form 1 4-^ 2 = -(A;TF4-2^), 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 right-hand 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 2-axis coincides with the positive M direction of the normal to the surface at x- M. and ?/-axes is z-axis, 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 direction-cosines 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 ) -, <^= 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 direction-cosines of r L with respect to the trihedral > T have the values (43) a _a a > Ib _ ^ s n ^ _ c cos c cos sin U + (b sin w u cos jj j r ^ o> s j n ^r ?7, M ) cos =6 [<7 f cos o> w -f c sin o) tt , where w u makes the angle which the positive direction of the z-axis 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?&> 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-> i; A tangent to the curve (46) Cv at M makes with the V-U=G. a>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 direction-cosines a v b { c l through These functions must satisfy the equations 3 0. , <? ; 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 , <? when b and c have the values ; 5 , ** ; 63 , <? 8, the expressions in parenthesis T ^>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 <W (y sin o> M z cos 2 cos W M ) sin &> 7, sin CT 1 (x z If in a y cos + (# M+ 2 sin Wu o> 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 L-y-. 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 direction-cosines 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 , <?, 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> + 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=<> ^=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 direction-cosines 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\ 5-5Ti 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 z-axis, we have, from (45), M P^VA^T cos (79) ) T, < ( /^ 1\ + sm ^ cos ^1 --, <I> \ds idco 4> r/ 1\ cos < 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 ((, T|T) 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>(p and from (47), and similarly Pd8=H \ A From du +p 1 dv), VA~C/> Qds=H^/~K^>(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<& \ as <&(p du +p +p l l dv) + sin cos < (q du + q l dv), (80) ds <I> (p du 1 dv) 4> (<? 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 rr-axis 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 zy-plane 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 x-axis 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 x-axis 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> which case the surface 5. minimal. When the lines of curvature are parametric, and the x-axis 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>, \PI PZ/ d< 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 x-axis 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 direction-cosines of the tangent to this curve are P <t>, <J>, ; sin * sin w, cos < 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 2-axis 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> ^ = i?-op, *?i=>?i-api- 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 = D-a, 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 > 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+l-i. 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> 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> 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 z-axis 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) ^-L-Jj. ~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, <& 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 <J> 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 2-axis of the trihedral T. Consequently . , /> 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 *"5FS-5 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 z-axis 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 zz-plane of the moving The con dition for this is Hence S2 is given by y\ ^L V i *!> ^i ^2 ^i / y<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/z-plane and zz-plane respectively of the moving trihe dral whose x-axis 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>, 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 < > 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> 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 =/>(!*, Conversely, two such equations admit three v), pendent integrals f^u, <* 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*> 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 =^ -- <& 2, v s are solutions of the equation ,. . , , ,. dudv Conversely, \^/p dudv : we have the theorem G-iven three particular integrals v^ 2 i> 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 direction-cosines of the normal to the surface. And direction-cosines 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> 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) = />,*), 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 one-parameter 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>+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. 332-367. 198 SYSTEMS OF CURVES is Lie has observed that the surface defined by (32) the mid-points 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 mid-points 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. Isothermal-conjugate 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 conjugate-imaginary 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 isothermal-conjugate. Evi an ana dently such a system bears to the second quadratic form lytical relation similar to that of * Vol. an isothermal-orthogonal system 107. I, p. ISOTHERMAL-CONJUGATE SYSTEMS to the first quadratic form. 199 that DD" EGF* be I) 2 > positive, In the latter case it was only necessary and the analogous requirement, namely by surfaces of positive curvature. Hence the theorems for isothermal-orthogonal systems ( 40, 41) are translated into theorems concerning isothermal-conjugate systems 0, is satisfied all by substituting In particular, Z>, 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 isothermal-conjugate system, isothermal-conjugate systems are given by the quantities u l and v l being defined by other real v1 u = const., u i+ where < 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= -!>", D =0. Hence the curves w const, and i\ = const, form a conjugate sys tem which may be called isothermal-conjugate. With each change of the parameters u and v of the asymptotic lines there is obtained a new isothermal-conjugate system. Hence if u and v are parame ters of an isothermal-conjugate system upon a surface of negative curvature, the parameters of all such systems are given by where < 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 isothermal-conjugate 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 isothermal-conjugate system. When equation (35) is of the form (33) or (34), we say that the parameters u and v are isothermal-conjugate. 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> 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 <> 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 f-c#=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>, 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, isothermal-conjugate, 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 my-i 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 isothermal-conjugate system on a surface, v be isothermal-conjugate parameters, is that satisfy (40); then the surface is unique to within its homothetics, , <^, that u and $ and its coordinates are given by quadratures. ,-< 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 , <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 mid-points 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 <>,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 isothermal-conjugate 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>, 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*/ \ Y-4- ds /V \ + ds ds T 7< 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 4-24du)\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<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/<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 = <f>(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 > we (43) <// 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 <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 = " <W W <W + -= ds ds - 206 The first GEODESICS integral of the second is 2 <// .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, <f> 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 <f> </> 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 one-to-one 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 <* _ 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 xy-plane in such a way that geodesies = with the linear element - v 2) du? + 2 w> 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. 80-86. 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. Jn-ft 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 = ^>(w), length of its arc is f Jo Since tegral G is > 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. 23-53. 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/>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. ^ EG-F*~~ EG-F*~ If, as usual, o> denotes the angle between these parametric lines, we have, from (III, 15, 16), -U v= n = U * > sin 2 ft) T? JF = COSft) . ? sin <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. 160-173. t Annali, Ser. 2, Vol. Ill (1869), pp. 269-293. 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 ___ FN-GM I 2 FM-EN 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 <f> denned by ._. (DO dc#> ) = ====================== EN-FM d<f> FN-GM ==: ? -=============================== 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 </> = 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 * < : When a one-parameter 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 </> 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> 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 <^ = 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 (/>.* 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/>, 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> 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 </> = 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<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> 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>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>(ui v, v )du. Furthermore, f (u)=f(u) + , ea>(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 <f> first and second deriva the left-hand member of this equation may be integrated by parts with the result " 1 wl-2- /& \v d , n ^lauaaO: du v d<l>\ LINES OF SHORTEST LENGTH for o> 221 &> 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> 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 <~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 <7 . M M , <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 < 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. 86-112; 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 > ^ 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 z-axis, 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< 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 orjual-conjugate representation of the surface is defined by MI sin I?!, z\ = upon a second surface x\ = MI cosi, y\ where ; <f>(ui) V CUi, C log Ui = - du where F denotes the function of M found by solving (i) for < . 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 xy-plane 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 near-by 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+/-+-A-1, u u u a 2 b 2 c 2 the parameter of the family and a, 6, c are constants, a 2 > b 2 > c 2 . For each value of (1) defines a quadric u, positive or negative, less than a 2 equation , which is 2 > ellipsoid when c u > oo, > > 2 u an hyperboloid of one sheet when b 2 u an Ian hyperboloid of two sheets when a > c*, > b 2 . As u zero. approaches Hence the smallest axis of the ellipsoid approaches 2 the surface u = c is the portion of the zy-plane, 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) < (u) = (a - u) (b - u) (c - u) - x 2 2 2 -y Since </> 2 2 (a -u) (c 2 2 - u) - z 2 < 2 2 (a - u) (c - u) - u) (b - u) = 0. 2 2 (b 2 2 (a ) < 0, c/> (b ) > 0, (c ) < 0, </> (, oo) > 0, the roots of equation (6), denoted by u l9 u the following intervals : , w 3 are contained in (7) a 2 > u, > b 2 , b 2 >u,> c 2 , c 2 > u^> 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 < (u) is u). equal to (u^u) (u 2 in (6) is u) (u s When by is (/> replaced this expression and u given successively the 2 2 2 values a b c we obtain * , , , FIG. 22 or = (8) = (*)(>-?) 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>u>b>v>c>0, an hyperboloid of one sheet when a>u>b>Q>c>v, an hyperboloid of two sheets when a>0>b>u>c>v. FUNDAMENTAL QUANTITIES 96. 229 culation (14) Fundamental quantities we find from (11) *. for central quadrics. By direct cal U ( U -V) _A ^_V(V-U) /() f(0) where for the sake of brevity we have put (15) = 4 (a - 6) (b - 0) (c - 6). : We derive also the following (a-b)(a(16) = and lobe \ ^ a)(c b) u v abc u v (17) JL> \ 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 />^ 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 > 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(b-c) 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 b-a 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) (a-b)(u-v) a (a-b)u-b][a(a-b)v-b] (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 (a-b)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. 121-132. 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<y \ Vo/ \V# V< 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, \ * > 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 twenty-fourth 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< of the values of u. Then at each point of these > 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 < . 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 <h i =- r A I (40) _ irr du - l\ 2J N(a 2J \(a-u)(u-c) 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) (u-b)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 &>, 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 ? ib-V\ = \U b/ We by take &> in place of t/r and indicate the relation between them ^fr =/(o>). 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 - *) [<* [ ^ ( > 1 *<H -1 find c) we make n % -yj- du -= d<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^) U-V of (43) \_\ |(c-o(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&> 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 <?, o>. replaced by i rr ~^~ du _i rr , ~^~ ^ (-*)<*-) 2 . * Journal de Mathematiques, Vol. XIII (1848), pp. 1-11. 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 (a-b)(b-c) 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 direction-cosines of the latter. The semi-diameter 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 semi-diameter 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. 556-686. 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>, 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 director-cone. 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 direction-cosines 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>, (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 direction-cosines of their common perpendicular. If the direction-cosines 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<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 near-by a2 loss sm2 6> -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 near-by 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> = 2 MM ton a- MM (7, In the limit M is the central point v tan<f>=lim ,, tan<f> (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 (/> 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 direction-cosines of the normal to the surface are of the form ( X=-(mz 2 nyQ ) + (mn m n) u ^ to ; the expressions for , direction-cosines 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 z-axis. To Then Let also the central plane be taken for the zz-plane 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 right-handed or left-handed 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> Cu<t>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 >*, 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. cross-ratio of four tangent planes to a ruled surface at points of a gen equal to the cross-ratio 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 + (u-a) 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> (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 cross-ratio. 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 ?<L?, /3 common perpendicular to two generators are z V* + < +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 z-axis 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 y-axis. MP. Find the asymptotic through the on the surface gener Show 6 that the equations where z = u-fwcos0, 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>, 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, + y-fz = 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. 354-357. 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 <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 <f> 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 <, ( 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 <f> 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 >, 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>=D"=c^=0, and equations (V, 27) reduce cu (>!")= from which it follows that <~/^= 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, > =0. From follows that PiP 2 where = -- = ~X Pi = |^| = |p 2 2 , /> 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>"=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<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 <, 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 ->/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 mid-points 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> 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 right-hand 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> terms of the right-hand 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>(v)dudv. normal We find for the expressions of the direction-cosines 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<r + ) > D= and of the asymp so that the equations of the lines of curvature totic lines are respectively (103) (104) 2F(u) du 3> (v) F(u) du 2 + <$> = 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 < 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 <& <& ^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 right-hand they are, of equations (98) are unaltered when u and v are replaced < F F 4> l/v and 1/u respectively. v) Hence the Cartesian coordinates differ at j of the points (u, and ( -- -> 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> , , 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 mid-points 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. 345-350. 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 <J>(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>"(v) + v$(v) -h iv( (109) iuf(u) + if(u) - -^- <l>"(v) = uf"(u) -) + v<f>"(v) 4> (v). It is clear that the surface so denned is real when / and <f> 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> 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>(v) ) where A, B, C are any constants whatever. Evidently, if / and are conjugate imaginaries, the same is not true in general of /,_ </> and <f> l ; but the surface was real for the former and consequently for the latter also. </> It is readily found that /t and x functions only in case J, J5, C are pure are conjugate imaginary is real imaginaries. surfaces. < 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 </>. follow the method suggested by Weierstrass.f establish first the following lemma We 4> : Gttven a function $*(? of , ; if in a certain 77, 4> 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 <E> : 4> = 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. 511-518. 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 ) +- < (f t irt, f + ti,] = 0, which proves the lemma. In applying this (101) that lemma to real minimal surfaces we note from l-Z ~Y~ to u l X = +v _u where u l-Z Y= _u 2i v, consequently the left-hand 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 <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> 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 direction-cosines of the normal are functions of x, y, Eliminating two of the latter between X _ > Y F(x, y, z) = _ _> 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 tion-cosines 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 < ) (v) dv, (112) {y = - ^ J/ & \ (1 +u 2 ) F(u) du i -I * dv. =i I uF(u) du I v<& (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. 563-567 (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"(Ydx-Xdy) Jv define a minimal surface which passes through C and admits at each point for tangent plane the plane through the point with direction-cosines 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 (Zdy-Ydz}\, 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 z-axis, and let the normal to the surface at a point of the line makes with the Y=sin<t>, x-axis, we have x = y = 0, z=t, JT=cos0, Z= 0. Hence the equations x of the surface are = - RiTsm <f>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 z-coordinates are equal, and the x- and ^-coordinates differ in sign. Hence Here is <#> : 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 direction-cosines 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). ^)> (|)"(c (0"<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 <i (a + c _ r ^ where r is the radius vector of the point (cf. 102). is 6. The tangent plane to the director-cone 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 near-by 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 <f> </> tf>, angle which the corresponding line of shortest distance pz> 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 >.* 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 -3a-2a = 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 z-axis, where a has the same meaning as in ; the envelope of the plane normal, at the mid-point, 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 = 1-2 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. W-SURFACES. 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-> 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 . > and consequently the surface is a sphere. CASE for z it II. c > a. From 2 the expression follows that sin > a < 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 < 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 < , ; > 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 + <?_ sinh - > 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 > 2 = \ sum2 - aw . , U , ; (8) (ii) v w = tfsinha 2= C \ 1 JN I <? a 2 .u -costf-du; 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 spool-like 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 hour-glasses. 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(/>). r = asin<, = 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/> c/> 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. 226-228. 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 </> V& = VG = (v) cos - + A/T (v) sin a , (12) < (v) cosh ci + i/r (v) sinh a , where < 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 isothermal-conjugate 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>. this constant is zero the surface is a sphere because of (13). this case, Excluding we 2 replace the above by a>, E= a 2 cosh F= 0, Now (16) D D n = a sinh a> a>. When these values are substituted in the Gauss equation (V, 12), namely 2 it is # I a^ L^ ^ H ^ HE o> 2 _ + du \ du ULE jv R found that must a ft> satisfy the equation 2 /18} a H--- + smh dv du o) 2 - a) cosh o> 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<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) &> 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 <"2 &> (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>c^ +cos 2 a>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 &>, 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 &> 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 direction-cosines 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>\ 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> + sin cos 9 du + sin cos 6 du &> sin 6 dv), sin 6 dv), o> sin # c?t> 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) <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. >S\, As by Ribaucour. (29) Moreover, if we put *-tan|. <. 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> 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 left-hand 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 left-hand 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 <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>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> du-\ (?M dv } / , \cosOdO X (cos ft) -f a sin&xi*; sin + o> 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 direction-cosines 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> sin 6 du 7 sin &> cos dv} X sin <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 <r ( --h v Q \ ) = a sin <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 cr-X2 =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>, dv/ 1 (35) smcr ( ) = cos sin <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 &> sin 0) &> du cos a cos C?M &> + a cos (cos co a sin cos cr (sin cos sin 0) dv, sin sin &) cos 0) &) + a sin cr(cosft) sin0o?v <* sin #(sin sin 6 + cos cr cos &) cos sin CD cosOdu), and the linear element of ^ 2 is 2 d** =a (cos <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 , <r 2 ), S by means of the respective pairs a function can be found without <f> quadratures which (<, such that by means of the pairs 2 f <r ((/>, 2) and o-j) the surfaces pseudospherical S and S surface S . respectively are transformable into a By hypothesis sin o-J is </> a solution of the equations * \H I sin <T 9( 2 */)= /p -^- 4- -^ = ^/l \ ) -- + sin $ cos l cos cr 2 cos (/> sin 0^ cos 6 sin 0. + cos cr 9 sin 6 cos ^, and also of the equations pi sin <7, p l* 4- - - = sn = d> cos 9 cos a. cos 6 sn (37) cos <> sn + coso-Sn) 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 <r 2 cos </>, a sin <r 2 sin (#>, ; 0. of The direction-cosines 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 <r 1 sin o>, sin sin <T I sin &), a), cos it cr 1 cos w, o-j cos w. From these and (38) follows that the coordinates of to <r M with respect to MM^ MQ^ l and the normal (j> S 2 are o-j a [sin a- + sin cr 2 cos [sin o1 a))], a [sin <r cos sin (<f> &))], sin 2 sin (</> 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 <r l sin & 2 cos (< (< &>) sin 0, sin 2 cos o^ sin <w), (39) l sin cr 1 + sin <7 6 l sin cr 2 cos ((/> + cos 0j sin sm & a If 2 cos o^ sin ($ <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 ((/> a l cos (# 2 0^ sin o- 2 ] cos sin a l cos cr 2 sin (# 2 0^ sin <7 co) = sin a (^> &>) l sin cr 2 cos (Q n #J, [sin -f- 2 sin cos (^ 2 cos 2 <7 0j) cr 1 siu crj cos 2 (</> o>) sin(^ ^)sin(^) w) = sin cr z sin a^ cos(# 9 ) 6^). Solving these equations with respect to sin and cos (</> (</> &>), we get sin ,. cos (<f> (0) sin =- sin -^ o-, o-j sin sin <7 2 cos (^ 2 o-- + (cos cos (0 2 /i <r. coscr 9 ^ - cr 1 sin <r 2 c/j) - -^ 2 X) + cos cos o-j 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> (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, <r, c), and that a particular integral ^i=/( v > *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 <7 ^ <r, l <f> 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 right-hand member of (42) order to handle this case we consider c in (41) to be a function of <r o-, reducing to cl for <r = a-^ L If the function tan for to Ism ~ we apply * the ordinary methods to which becomes indeterminate a <r, = o- numerator and denominator with respect v differentiating we have or tan /6-w\ = 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 isothermal-conjugate system of lines. Since the parameters in terms of which the surface is defined in 119 are isothermal-conjugate, 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 &> = <(#, 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>(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 <r er Q sin (9 . + x <). *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>), dp (6 &>) = sin (6 + &> o>). 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 TF-SURFACES 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 <r <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. <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<? Xi + coswJT, X X are direction-cosines, with respect to the x-axis, 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. W-surfaces. 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 W-surfaces. Since the prin cipal radii of surfaces of revolution and of the general helicoids are functions of a single parameter ( 46, 62), these are TF-surfaces. shall find other surfaces of this kind, but now we consider We the properties which are common to TF-surfaces. 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 /> 2, as the integration of equations (45) r dpi is reducible to quadratures, thus : =Ue J *-*, Crelle, Vol. V^= Ve r J Pl </p 2 ~ P2 , LXII (1863), pp. 160-173. 292 JF-SUKFACES 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 * =* / <*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, </> When these values are substituted in the Gauss equation for the sphere (V, 24), the latter becomes 1/* du \ </>" M du) , jL/* dv \tc* aY dv) _1. = K<f> but This equation places a restriction upon the forms of K and it is the only restriction, for the Codazzi equations (45) are <(), 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 W-surface 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 JF-surface 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 right-hand 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 < pi Ve = p^ "- ft , vG = p,e _ r Jf Pi -p >. 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 JF-surfaces 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 W-surface can be found by quadratures. 294 JF-SURFACES The evolute of a JF-surface pos 124. Evolute of a W-surface. 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 JF-surface 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 W-surface 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 W-surfaces; the relation between the if Sl be deformed in any manner whatever, radii of these W-surfaces 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 JF-SUBFACE 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 W-surface ; 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 TF-surface, 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 TF-SURFACES 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 &>, and replace u by au, the linear da-- element (60) becomes 2 = sinlr co du + cosh 2 o> 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) <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 <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 > (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 IF-surfaces 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 ., < = .to sin-> . =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 mid-points 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 IP-surface of the M M RULED JF-SURFACES 126. 299 Ruled W-surfaces. We conclude the present study of Tr-surfaces with the solution of the problem : To determine the W-surfaces 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 ! u-a 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 + <?, 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. 205-210. 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 W-surface 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 TT-surface for which (i) P-2 - PI = COt - 4. The const, helicoids are the only >F-surfaces 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 />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 z-axis 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 z-axis 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 -- - > 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 direction-cosines 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 -> 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). > 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)Vl-a 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 one-parameter 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 director-developable, 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 one-parameter 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 director-developable. 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 director-developable 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>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 director-cylinder 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>, 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</ , 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. 181-192. 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 direction-cosines 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 one-parameter 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. 200-210. 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 9-0, 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 direction-cosines 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 direction-cosines 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 < 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 one-parameter 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 , />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 ic-axis, or by a line perpendicular to 318 TF-SURFACES 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 + u-dv 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>, 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<r that the general integral of the corresponding equations <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 2-axis. The tangents to a family of geodesies of the elliptic or hyperbolic type ; on a pseudospherical surface are normal to a W-surface 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" >F-surfaces 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 JF-surface 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 IF-surface 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 m-dv. 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 director-developable 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 director-developable and a second surface of Monge, which has the same director-developable 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 TF-SUKFACES 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> F t> = 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 one-to-one 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. 371-387. 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 left-hand 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 + />, 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 (/> (U, V) = $(U J , V 1 ), ^ (U, V) = ^<(U , V ), establishing a one-to-one 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;</> is A^ = , A^, f)=A!(<#> , 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 (/> = const., ^r const. ; </> = 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 < A > 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. <r * If the surface be referred to the curves where a that Vol. AI(<T, J ~vf(K) &<<?) C= = = const, and their orthogonal trajectories, , equation , A%<r (6) may be replaced by A2 tr = A^ , and it can be shown Cf. Ai(<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 =*<*). 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), > <f>(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<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 2-axis, 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<& (v) F(u) value of these 4>(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 <?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 <+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 z-axis, 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<z , z-axis. 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> 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 z-axis 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 one-to-one 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<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(w-t> ) -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^r-sr) J dir 1 T-T- Tl and therefore is of the form oTi cv 0tr \dudv / \_du studied by Ampere. Hence we have the theorem: will give the ^<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 G-iven 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 left-hand 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 right-hand 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 < 1. Hence we have the theorem : If z be a solution of A 22 (1 A X 0) K such that A^ < 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 > 1 V<? dx - 1 c>f 1 the values of to r-i = cz \IG co vV; w tor v = U, for v = 0, cu cu du the latter being the direction-cosines 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 ; > 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 />/>" /> 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> must have the same value for all surfaces, for all the surfaces. If o> 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> has the same value, unless w is a right angle. Hence it is have two applicable surfaces passing through a curve whose points are self-correspondent, 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> 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. ($>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) *<f>(u) <S?-S<f+ <#>(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 ;>I = K+ 2 2 =<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 *a-a& = 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) rf-rf-li ^-tf+lq* The first of (28) reduces to pi we introduce a function thus o>, = l. In conformity with this p = cosh l &>, q = sinh o>. Then equations (29) may be replaced by ft), = sinh p[ ^ = cosh o>. 338 DEFORMATION OF SURFACES Moreover, the fundamental equations (24) reduce to _ dv a 2 o> du V] , o> du , : a H c -- = o> o2 smh . , cosh o>. 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 <f>(u)=p[<t> 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/ />( 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 > 1-22-1 .D f!2V are firi-D". /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//> we have 2 du Now equations (33) are equivalent to (34), and & log p r,<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 (/> ^ : 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</> tions (34) reduce to f 12"| 1 \ ~\ a ** > | ^ 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 <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 < (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>", 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 </> 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 direction-cosines 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 +jtf-K -!. 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 director-cone 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 < = cos < m = cos sin i/r, = sin </>, which evidently satisfy the second of reduces to 2 (39). The first of (41) (43) If <J> +^ 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 < 2 ], and analogous expressions for y[ and z[. trary function of v, and ^ be given by (46) Hence, be an arbi +=(^f^ J COS (/> 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 director-cone, 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 (/> /(c/>, -<fr) may /(<, take ifr) / 0, by combining equations (43) and and ^r as functions we obtain the expressions for as arbitrary; for, < 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. 297-302. 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 director-cone. 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 zy-plane. 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/>.( (43), 2 reduces to 2 <// +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 direction-cosines 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. <r (46) I =# cos # * -f sin (/ cos <r +X sin <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 direction-cosines 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> 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 a-2a* -2V2a .2-a-2a* 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>(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^:. ^L-Q^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)<f> (v) + i//(v), z = /(u)[0(i>) (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 x-axis parallel to the yz-plane, 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>i, $2, ^i, ^2 satisfy the conditions $2 + <I>| = 02 + 0| - K, 4>i* + <J> 2 2 = 0{2 + 02, ~ $1(0212 Show also that the determination of 4>i and 3> 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 direction-cosines of the curve in which the director-cone 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. 159-200. 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 z-axis 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 zy-plane of the moving trihedral, we must have (3) rdu if + r l dv = Q. Hence the a trihedral ner, as the vertex of 2>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>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 /// <f> It") *+*<> 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 z-axis 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 right-hand 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 direction-cosines of the axes of the moving trihedral with ; , Zj> respect to fixed axes, we have, from (V, 47), du (9) il dv " _ Xa qi q<> - 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 z-axis 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 z-axis 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 *>)] + [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 + >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 <% __ ^_ -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 (/> 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 direction-cosines 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 = wX-f-X. 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^,2-r ^-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> 361 <fu =a ^ 1 dp + , 3 -i<t> , dt> = tf$ i- , <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 direction-cosines (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<f> dp the principal radii of s v= d(f> * 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>, 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 z-axis 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 direction-cosines of the normals .A., 2 ~ Zs* are 1 fo Y = -- > 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 </>(>, #)=/(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 < (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 <j> ^ = 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/3-1 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 <j>(p, ; (64) In this case so that equation (37) reduces to (65) />i + ft = -(2;> + 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> 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 *> . (t>)] 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(l-m)p\ m being any constant, equation (72) becomes m(l-m)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(m-I)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 > 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> = (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 z-axis 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, > 0, or < 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, = > < 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, sc-axis of the where X\, JT2 , X are the direction-cosines 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, direction-cosines 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< 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>(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 <S m z is positive or nega x + iy = v, x-iy = 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 direction-cosines 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. 296-310. CHARACTERISTIC FUNCTION Excluding the case where S <r-\ 375 is a developable surface, * we solve these dx. equations for >,^ *-l du 2*X-zr 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 <. 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 right-hand 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 <", 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 <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> 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 , >?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> Z>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 <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< 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> 379 When (17) are the second fundamental quantities for $. these values are substituted in (15), we find " , Z> , D X-r=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 <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 - <rZ> = pl>" 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 <> which are consistent only when that is, when the point equation of S, namely j^ dudv fi21<tf \l J cu \ZJfo ri2|<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 > 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 /* <r <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 left-hand 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 right-hand 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 direction-cosines 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 direction-cosines 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> .*" = .ZX J>, 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 <r p p (ED* FD), X == p (ED 1 GD), =?<p (FD" GD f ) satisfy equations (19).f equations of the form Upon S-s- 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 near-by 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>"\ 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. <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 <zi, 2 =x -r} y tyi, f2 = z tei, where 2. any constant. two surfaces are applicable, the locus of the mid-point 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<f>* , , = -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 isothermal-conjugate, and S Q and S Q are associates of one another. <r /j. <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>), y=Vb(u-v), z = 2uv, ; the associate surfaces are surfaces of translation whose generators are plane curves their equations are x = Va(U + V), y =Vb(V-U), 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. <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, xi-xz = and that the 17. yzo - zy , yi-y 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 <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} > y = fz( u v -> z )-> jz( u v ) -> > / and < under consideration, and the functions are analytic in the domain of are such that (/> u and v * Crelle, Vol. LVII (1860), pp. 189-230. 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 <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 direction-cosi 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 , *>dX\-, { <?- 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 direction-cosines 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 near-by 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 right-hand of this equation, the result is reducible to <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), $, jC-J.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 right-hand member with respect to we get a quadratic in t. Since $&*> 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 near-by 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 <o ^9X, du ^dX. dv & (22) so that the direction-cosines 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 direction-cosines . (22) given by cos &> = 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 &> -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 (<&!> (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 mid-points 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 &> 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> 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 direction-cosines of the normal to a minimal surface at the point (cc, T/, z), the line whose direction-cosines 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 - (/ + )<^+ 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 direction-cosines Xv Yv Z l satisfy the conditions u i) where normal and 4>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) <*> 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 one-parameter 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+\-* fc-jr 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. 16-22. 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 * > 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 ~~~ c-c/ * 7 c/ =^.A 01^ > (45) foo/ Ctf CA M* fl (Q^/ CV^ *>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<o, g = r& first When these values are substituted in (47), the two become 12, (49) f? a/ / \ 7 = ^ 1 d i , (r < 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\4-2f * + 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. 342-344. 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 direction-cosines 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^ ^ _ /}?> 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 5-i 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) < = 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 + <^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 - + /><^=0, the function W> (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 <F=^=1, c? = COSQ), where co is a solution of = sin dudv In this case equation (53) (65) is ft). - = p cos <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. />, 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. 95-102. *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. 161-172; also Lezioni, Vol. I, *Annali, Ser. 2, Vol. XV (1887), pp. 323, 324. JF-CONGRUENCES 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. TF-congruences. 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 W-congruences. erties of these congruences. JF-surface (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 JF-congruence is and sufficient condition In order to obtain an idea of the analytical problem involved in the determination of TF-congruences, 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. 126-127. 418 RECTILINEAR CONGRUENCES y, z, and similar equations in vv i> 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 />) = ^(x - x)*= 2 7?i 2(^ =w 3 a 2) iy> 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> i/j, 2, i> 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 JF-CONGRUENCES 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< 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 W-congruence. 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 direction-cosines of the generatrices of , so that we have : focal surface of a W-congruence 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 IF-congruences for which the sphere one of the focal sheets. 2. Let vi =fi(u) + 0i (w), where /; and - </> 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 direction-cosines 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 direction-cosines are of the form 8uBv (1-f-ww) 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 mid-points 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 IT-congruence 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 TF-surface 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 two-parameter systems of straight lines, but is * applied to two-parameter 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. 1-10; also Eisenhart, Congruences of Curves, Transactions of the Amer. Soc., Vol. IV (1903), pp. 470-488. 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-> T;, ^ and the rotations p^ qv , r^ satisfy the conditions dp_di_ (3) __ d|__Mi = dr dr. ^Wl (1) PH Ph-M^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) direction-cosines 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* > 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 cross-ratio 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 <j> (16) 24>=,K -a -& 430 CYCLIC SYSTEMS must </> 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) < of the form du -r ^_r du dv 7, _(__r_] -\-J^ \dudv / +L ducv <>, + 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 x-axis 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 z-axis 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, /> 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 mid-point of the line of the congruence, we find that We (21) replace this equation by the two 8 p cos a-. <r, R p sin cr, thus defining a function /5 1 Now we have 1), =/E)(coso-+l), /? 2 =/o(cos<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 (coso--l)]=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 mid-point 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 2-axis 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 mid-point 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) + py-qx = 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 <,=>-* <#> + *, +^ 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 direction-cosines 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 + <?! 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 <r ( ,1 cos a 1 (34) , Cdu cos <r sin sin <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 ), ./ ! l-fCOSCT\ 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 &> = 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 &> 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> fi sin &> o> f) //, du &> cos (R sin -f- cos 77^ dv 0. Hence the (41) quantities in parentheses are zero, from which i; we o>. obtain R= cos &) -h T] I sin w, ^ = f sin o> + 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 &>, 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 < 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<r*=sm*0du*+ cos 2 <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 zy-plane, as before, that of the circle, and take the z-axis 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 rfx-ftf = 0. are functions of </> 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. 127-132. 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 > B-V = -* 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>"(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 <f> (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 z-axis is parallel to the axis of the circle under consideration. In this case equations (14) may (46) be replaced by 2 A> = 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 <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 <r +1) - S-L. 2cos<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 z-axis 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 two-parameter 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 -<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> 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 x-axis tangent to the curve u = = const. 0, The equations z x = p(l + cos0), y = />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 two-parameter 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 two-parameter 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 y-axes respectively. Show that ff denotes the angle which the radius MMi makes with the x-axis of the trihedral, the lines of curvature on Si are given by sin <r (sin <rp r cos <r) du 2 + in[qi \ I sin <r ) dv 2 dv/ H fllfl -) (cos <rri + p sin <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 <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> + V6? sin o>) sin <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, <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 direction-cosines of the normals to the H surfaces u i = const, we have (5) , du. We (6) choose the axes such that = 4-1. 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> 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. 73-79. 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 < 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> 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) l-l^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> a = cos a o) du* + sin a o> dv\ where is a solution of the equation d a o> d a o> _ sin o> 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>vow 1/f^, where (188,",), ?/8 is a function of ?/ alone. 2, Vol. XIII pp. 177-234; Vol. et les XIV (1880), pp. 115-130; Lezioni, chap, xxvii. wr les ni/stemes orthogonaux coonlonntes curvilignes, pp. 308-323. Paris, 18U8. TRIFLE SYSTEMS OF BIANCHI In accordance with (11) and (18) _1 P* , 453 we put tan<w 1 dff, (19) 1 g// cot <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> = do) cot a) From (20) these equations we < have, by integration, cos co, ffl = 13 H= z </>., 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 <. I (21) / // tan o> [ cot o> \ A d da) --- - log z ,5w, ^3 \- \ ! / From these equations follows that Hence, unless l;l and ^>., ;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> ?< . 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-- \ * -w-r-ft . (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*+<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> When exception is made of this case, the functions . 13 and </> 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< (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 <J> But by changing the param affect the , form of (26), we can give (28 ) <& 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 near-by 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> cos w 3 last _ 8<a u> = dot cos u sin u> 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 < = 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 zz-plane, 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 direction-cosines of the tangent to the <, 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<f>\ _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. 330-333. 458 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES that (f>\ The condition d_ /i ?1 it be cyclic d_ is sin cu \ (30) R cos (/ / dv\ R , sin <f> d /sin (f) \ d icos<f) \_ /> 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 two-parameter 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 <j) we have, by (11), 1 <f> 1 d//., sin dH z and the equations analogous 1 to (31) are 1 />, 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. 6-8. 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 f-f ^-\ ^ 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>), 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 direction-cosines 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 tion-cosines 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. 102-122. 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 < > $-* 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 < > *--/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 <t>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 <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 one-parameter 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 cross-ratio 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^ + <* 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= N-VU 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, 280-283, 290, 318, 320, 370, 456 surfaces of, 370, 371, 442, 443, 445 generalized ; ; two given surfaces are applicable, 321-326 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, 378-381; 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, 189-194 orthogonal, 129 spherical represen straight, 140, 234 tation, 144, 191-193; preserved by protective transformation, 202 pre served in a deformation, 342-347 ; ; ; ; transformation of, 439 triply orthog onal systems of, 452-454, 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, 284-290 Beltrami (differential parameters), 88, 90; (geodesic curvature), 183; (ruled W-surfaces), 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, 59-61 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, 39-41 on a ruled surface, 250 ; deformation, 348 Bianchi (theorem of permutability), 286-288 (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, 374-377, 390 Corresponding conjugate systems, 130 Cosserat (infinitesimal deformation), 380, 385 Cross-ratio, of four solutions of a Riccati equation, 26 of points of intersection of four-curved 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, 155-157, 161, 168, 170, 189, 200 Combescure transformation, of curves, of triple systems, 401-465 surface, 184, 185, 283, 290, 370, 464 Conforinal representation, of two sur faces, 98-100, 391; of a surface and its spherical representation, 143 of a surface upon itself, 101-103 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 Conformal-con 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, 401-403, 411; ruled surfaces, 393, 398, 401 limit points, 396 prin cipal surfaces, 396-398, 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, 421-424 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, 409-411, 412, 414, 416, 420 derived, 403-405, 411, 412 isotropic, 412, 413, 416; of Guichard, 414,415,417,422,442 pseudospherical, 184, 415, 416, 464 W-, 417-420, 422, 424 of Ribaucour, 420-422, 424, 425, 435, 442, 443 mean ruled surfaces, 422, 423, 425 cyclic, 431-445 spher ical representation of cyclic, 432-433 cyclic of Ribaucour, 435, 442, 443 developables of cyclic, 437, 441 ; ; ; ; ; ; ; ; Cyclic system, 426-445 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, 312-314, 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, 338-342, 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)", ; &"> 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, 91-93 tangential, 163, 194, ; ; ; ; 201; elliptic, 227 INDEX deformation), 332 (surfaces appli cable to paraboloids), 367 (triply ; 469 ; ; orthogonal systems), 458-461 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, 327-330 of surfaces of constant curvature, 321323 general problem, 331-333 which changes a curve on the surface into a given curve in space, 333-336 which preserves asymptotic lines, 336, 342, 343 which preserves lines of curva which preserves ture, 336-338, 341 conjugate systems, 338-342, 349, 350, 443 of ruled surfaces, 343-348, 350, surfaces) (see ; ; Element, of are.a, 75, 145 linear (see Linear element) normal sec Ellipsoid, equations, 228 tion, 234 polar geodesic system, 236-238; 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, 353-369 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, 61-63, 64, 69, 442 of a one-parameter fam ily of spheres, 66-69 of a two-param 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, ; ; 84-88, 90, 91, 120, 160, 166, 186 of the second order, 88-91, 160, 165, 166, 186 Diui (spherical representation of asymp totic lines), 192; (surf aces of Liouville), 214 (ruled TF-surfaces), 299 Dini, surface of, 291, 318 Director-cone of a ruled surface, 141 Director-developable 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, 45-47 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, 124-126, 129, 150 cyclide of (see Cyclide) theorem of Malus and, 403 ; ; jE, &> F, G, definition, 70 for the moving trihedral, 174 definition, 141 for the moving trihedral, 174 e,/,/, flr, definition, 393 ; &> ^ ; Edge of regression, 43, 60, 69 * Family, one-parameter, of surfaces, 59, 446, 447, 451, 452, 457-461; of planes, 61-64, 69, 442, 463 of spheres, 66-69, 309, 319, 463 of curves, 78-80 of geo desies, 216, 221 Family, two-parameter, 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, 409-411 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, 146-148 parameter meridian of, 140 geodesies, of, 140 surfaces of center of, 151, 209 149, 186 pseudospherical, 291 is a IP-sur ; ; ; of a curve, 18 Frenet-Serret 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, 228-230; 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, 207-209, 230, 276 Geodesic representation, 225, 317 Geodesic torsion, 137-140, 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, 385-387 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 TF-congruence, 420 Intrinsic equations of a curve, 23, 29, ; ; ; ; ; 30, 30 ; ; ; Invariants, differential, 85-90 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, 43-45, 311 of a surface, 180, 184, 300 Isometric parameters. See Isothermic ; parameters ,- * Geodesies, definition, 133 plane, 140 equations of, 204, 205, 215-219; on surfaces of negative curvature, 211 on surfaces of Liouville, 218, 219 Goursat, surfaces of, 306, 372 ; ; ; Isometric representation, 100, 113 Isothermal-conjugate systems of curves, 198-200; 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, 93-98, 209, 252, 254 formed of lines of curva ture (see Isothermic surface) Isothermic parameters, 93-97, 102 Isothermic surface, 108, 159, 232, 253, 269, 297, 387-389, 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 JF-surfaces), 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, 421-424 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, 254-265, 318, 391 on a sphere, 81, 257, 364-366 ; ; 390 ; Minimal straight lines, 48, 49, 260 Minimal surface, definition, 129, 251; asymptotic lines, 129, 186, 195, 254, 257, 269 spherical representation, 143, 251-254; ruled, 148; helicoidal, of revolution, 160 149, 330, 331 parallel plane sections of, 160 mini mal lines, 177, 186, 254-265; lines of curvature, 186, 253, 257, 264, 269 double, 258-260 algebraic, 260-262 evolute, 260, 372 adjoint, 254, 263, ; ; ; ; ; ; ; ; ; ; ; ; ; tion of, 143, 148, 150; osculating plane, 148; plane, 149, 150, 201, 305-314, 319, 320, 463 plane in both systems, 269, 300-304, 319, 320 spherical, 149, 314-317, 319, 320, 465 circular, 149, 310-314, 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, 327-329, 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, 305-308, 319 Moving trihedral for a curve, 30-33 applications of, 33-36, 39, 40, 64-68 Moving trihedral for a surface, 166-170 Monge ; ; ; Loxodromic curve, 140, 209 78, 108, 112, 120, 131, rotationsof, 169; applications of, 171183, 281-288, 336-338, 352-364, 426-442 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, 80-82, 91, 119, 129, 177, 187; par ametric, 75, 93, 122, 134 geodesies, 187 isothermic (see Iso thermic) Orthogonal trajectories, of a one-param 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, 275-277, 283, 317, 318 mation, 277, 323 transformations of, 280-290, 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, 452-456, 464 ; ; ; ; deformation of, 348, 349, 367-369, 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, 234-236, 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; conformal-con 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, 420-422, 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 director-cone, 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, 270-272 geodesies, 275-279, 318 deformation, 323 lines of curvature, 278 276, invo transformation, 278-280, 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, 343-348, 350, 367; spher ical indicatrix of, 351; infinitesimal of a congruence, deformation, 381 393-395, 398, 401, 422, 423. See Right conoid, Hyperboloid, Paraboloid ; ; ; ; ; ; ; ; ; Scheffers (equations of a curve), 28 Scherk, surface of, 260 Schwarz, formulas of, 264-267, 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 ; ; 109-111 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, 314-316, 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, 412-414, 422, 432-435, 437, 441 Spherical representation of a surface, fundamental quan definition, 141 lines tities, 141-143, 160-165, 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, 305-308 Surface of constant mean curvature, definition, 179 parallels to, 179 lines of curvature, 296-298 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, 326-331, 341, 349-350, 362-364, 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 >F-congruence, ; 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, 53-55, 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, 447-451 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, 41-44, 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, 137-140, 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 W-congruence, 417-420, 422, 424 fundamental >F-surface, definition, 291 quantities, 291-293 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; (>F-surfaces), 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, 353-372 triple system of, 455, 450, 403, 464 ; * For references such Transformation of BJickluud, see Bitcklund. 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