and met the waggon just40 minutes before he came to the
thirty first milestone. How far used to be B from L when A reached it?
Fifty eight EQUATION PAPERS.
4.
1. I(5x-l)- A"(7x-2) = 6J-stock of corn, expectingto sell it in
6 months at 3s a bushel more tha

335. There are a n* of tickets,marked with some some of the bneumr-s
from 1 to n'+ 1. Each this sort of numbers, (such
as r,)is marked upon the same quantity (r) of tickets ; and
every ticket marked with a square number m^) confers a
prizeof m shillingsA.

x (g,fifty four MISCELLANEOUS EXAMPLES.
344. If tty h,Cf "c. Are any n portions, shew that "+ 5*+c"+ "c
"n aho .);and thence show that 1.2.3 .,n "^ (n+ l)
345. A playsat a game, wherein he reckons his risk of success
to be e. If it be a fair danger that h

1* 2" 3*
^
!" 2" three"
^
!" 2" 3"
^ 2+2-^+23^-' 3
+ 3-i+33 + "c., ^-^, + _-"c.
328. What number of triangular pyramidscan be fashioned, whose edges
are six givenlines,any two of which can be " than the 1/3 ?
329. Do away with m, w, jt;,q from the equatio

819. Investigattehe generalforms of x and y, which rationalize
ax^ 4- hy*^ywhen w = 3 or any bizarre quantity.
320. Fifteen folks sit down at a round desk. Shew that it's
6 to 1 againsttwo particulaprersons sittinngext every different;
and that,generallyf

a ^a-y)x^(a - 2y) a:*four- "c. Ad inf, *
and if x obtain values in h.P., shew that the corresponding
values of y will likely be in A. P.
X^ x"
259. Shew that the series x - " " " + - "c. Converges
at the w^h time period, if n"x and find.
260. There are th

308. If z" + a + 1 = zero, shew that the sum of those phrases of the
expansionof (1 f a;)", in which the index of a; is a mult, of 3,
= J (1+ xzf + (1 + xy four- (1 + xz'r.
2 three four
309. Sum to n phrases and ad infinitumr-^-r + ^-77:
+
^"^seventy seve

2d half a consistent ratio. Shew that there is however one
such sequence.
245. Compare the chances of throwingtwo aces onlyin two trials
with three dice and in three trialswith 4.
246. If 2a;,= y^'*^"=' - y^'-^'t^h'en, x, (x^+ a;,+ "c. + x^,^,=)x"*,
247.

coin is taken from one (it's not identified which) and
droppedinto the other; and then, on drawing a coin from
every bag, theyare observed to be two shillings.What's the
chance that this may occasionally arise once more,if two extra are drawn,
one from ea

ax*+ with the aid of*+ cz'=)(a^3 + 6^+ c^)ok^.
232. Shew that,if pq = r, the equationa^ + px^ + ja; + r = 0 will
have two roots equaland of oppositesigns.
233. In finding three numbers with primeden", whose sum shallbe 1 ifs*
234. Resolve into partial fra

219. Sum to n terms
220. A
J tossinga coin,is to pay B is if it fall heads the firsttime,
25 if the 2d,3s if the 0.33,etc for n throws, the
sport to cease as quickly as itfallsheads. In finding ^*s expectation,
221. Get rid of a;,y, z, from the equations

207. Do away with a and h from the equations
208. S^yS^,S^y"c. Are the sums, to n phrases, of n Geom. Sequence^
whose firstterms are every team spirit,and fashioned ratios,1,2,three,
"c.: shew that
^j+ ^,+ 2^3+ three/S;+ "c. + (w-l)^"=r-i-2"+ three" + "c.

192. K flj: cfg: : a, : "three : : flg: cf^"c., then
("!*+ fl,+*"C.)(rtg+*^three*+ "C.)= ("!",+ rtgfl+fj"c.)'.
193. Shew that (1+ xj'+ (I - xf)v (1+ a:)+ V(l + a:)= 1 - a long way,
nearly,when a: is small: and to find its approximate value
when X is Zar^e

176. LiA^B, J^x^C, and Coc^Z^+^ZB*, shew that
177. Shew that abc"2a"h)1h"c)2c"a), except "=i=c.
178. A bequeathsto h"fk1st baby an n^ of his property +"P, to
the 2nd an v-^of th"the rest +"2P, and many others. They all
share alike : how many have been the

132. Symbolize^/2n^-l) in the form of a binomial surdj and
shew also that V(four+ three V- 20) + V(4 - 3 ^/- 20) = 6.
133. Given log If = .1461280, log.A hundred and forty four = 1.1583625,
and log.0441 = 2.6444386, in finding the logsof the 9 digits.
134

145. Divide unityinto 4 partsin A. P., so that the sum of their
cubes may be ^q,
146. Shew that (aJ),+ a,+ "c.)'" (a*-fa* + "c.)b*+ b* + "c.),
unless flj: tj= a^ : b^- "c.
147. Find the coefiicientof x* in (a + 5ar+ ex*)c"*.
148. Given A, my sales, a the

and what should J? Have staked, that -four's drawing first
could givehim no competencies?
121. In finding the n" of divisors of a hundred and forty, and the n" of numbers less
than a hundred and forty and high to it: express generallythe rational
values o

a^-x- V("^ - ^ ) ^ c^ a^- a*
110. The veracities of A, B, C, being f, J, f, C asserts that
i've gained a prizeof "10 in a specific rafile,where there
are 10 tickets,of which I preserve two: what's the value of my
expectationi,f I in finding that C has onl

squares of the terms = productof sum of all the phrases through
excess of the atypical terms above the even.
Ninety six. If the G. Imply between x and y : the H. Imply :im :n, then
X : y i: m + v'(m*- w*): m - V(m*- w*).
97. Find the sum of the entire num

Y
,
^
. 80. A had in his pocket a sovereignand 4 shilling;staking
out two cash at random, he promisesto provide them to B and C:
what's the valued at of C"s expectation?
Eighty one. Shew that (" + " - cf + (6 -f c - ctf+ (c + a - 6)'" Zabc.
Eighty two. El

be the Arithmetic mean between m and w, and the Geometric
mean between a and b.
MISCELLANEOUS EXAMPLES. 35
66. Find the coefficients of x'^ and x* in (a - 5x* "]"ex - dx~^Y"
sixty seven. The mail from A begins for B at p hrs P.M., and that from B
for A oi

(^a+b+c+df = (a+")H(c+e^2^2(5+c)2.
381. The No. Of Var"^ of n thingsr,together: the No., r-1 gteo-ther
: : 10 : 1,and the correspondinNgos. Of Comb"^ are as
5:three; find n and r.
382. A character makes 20 lbs. Of tea at 4s. 9J.,with the aid of mixing thr

taken from each to make 5 gals,of wine and 9 of water ?
18. Obtain the square root of 1 + (1- c*)"* in the form a* + /3^19. Solve the equation H = -.
A + a: *J(a^ x) x
20. A flag-sta(fc)fstands at the top of a tower, whose heightis a.
Find the distance fr

= 2 (a- 6) (rt- c) + (6 - a) (6 - c) + (c - a) (c- 6).
4. A starts from a certain place,and travels a miles the first day,
2a the second, 3a the third, "c.; after 4 days, B starts to
overtake him, travelling9a miles per day. After how many
days will he co

and if 2^. Be taken from ^'s reward sum and added to j5's,
the latter amount will probably be of the previous. What had they
every at first?
367. To find the worth of s/a-6axndysquare the outcomes.
368. If the differenceof two fractions be mn- show that 7

end sixteensooner than he otherwise would : what time
will he take,if he onlybeginto quickenhis % halfway?
E349. Divide (^-l)a"-(:r3+^2_2)a24-(4A^H3.'2r)-af-3(^+l)
through Or-lK-(^-l)a-j-3.
350. MultiplyA^a'^ VKc)Hya/"^-v/(A)2.
351. If x=^i/'-ir-'Vr'-ijq^

A " n.A"ln a " Sn,^
1 ^"c.
Nnn
320. If the sum or change of two numbers be 1,exhibit that the
change of their squares is the change or sum of the
numbers respectively.
321. A servant agreedto reside along with his master for "S a 12 months and
a livery,bu

numbers is divisibleby three times the middle number.
284. If a : 6 : : c : e7,show that ^a" - W : 2^ ~ 3^?^: : aH^' : cH^285. Two-thirds of a designated quantity of bad folks obtained
Is. Qd, each and every,and the rest 2s. 6c?. Each and every : the whol