531] 
A “ SMITH’S PRIZE ” PAPER ; SOLUTIONS. 
457 
Hence 
But 
Hence 
4$ = (a' 2 + b' 2 — a 2 — b 2 ) sin 2t + 2 (aa! + bb r ) (cos 2t — 1) 
= (a' 2 + b' 2 ) sin 21 — (a 2 + b 2 ) sin 2t — 4 (aa' + bb') sin 2 1. 
a! sin t = x — a cos t, 
b' sin t = y — b cos t. 
(aa' + bb') sin t = ax + by — (a 2 + b 2 ) cos t, 
(a' 2 + b' 2 ) sin 2 1 = x 2 + y 2 — 2 (ax + by) cos t + (a 2 + b 2 ) cos 2 1, 
and substituting these values 
2 sin t cos t 
4 S- . 
sm 2 1 
— 4 sin t 
= 2 (x 2 + y 2 ) 
or, what is the same thing, 
cc 2 -\-y 2 — 2 (ax + by) cos t+ (a 2 + b 2 ) cos 2 1) 
— (a 2 + b 2 ) 2 sin t cos t 
{ 
cos t 
sin t 
ax + by 
(a 2 + b 2 ) cos t] 
cos t 
sm t 
(ax + by) +2 (a 2 + b 2 ) 
sin t ’ 
8 = | (¿c 2 + y 2 + a 2 + b 2 ) cot t — 2 (ax + by) cosec t, 
which value of S satisfies the two partial differential equations. 
More generally, the two equations are satisfied by 
S = c + foregoing value, 
(c an arbitrary constant) which new value, considered as a solution of the first equation, 
contains the three arbitrary constants c, a, b, and is thus a complete solution; and 
similarly considered as a solution of the second equation, it contains the three arbitrary 
constants c, x, y, and is thus a complete solution. 
I venture to add a few remarks in illustration of what is required in the papers 
sent up in an Examination. 
In the latter part of question (2) (form of the equation for a system of parallel 
^ l^/V^Yl 4 [p . «I] 
must be constant: 
ffV 
\dx) 
+ 
curves) it is worse than useless to say , . ,. , . , , , 
CtG I\CvOGJ yCby > 
a good and sufficient answer would be that it must be constant in virtue of the given 
equation f(x, y, c) = 0. So in question (13) (the Lagrangian equations of motion), it is 
quite essential to explain [p. 455] that T, U are given functions of £, v, •••.• V, ••• and of 
y,... respectively; but for this the equations might be partial differential equations 
for the determination of T, U, or nobody knows what: it is natural and proper to 
explain further that T is homogeneous of the second order in regard to the derived 
functions £', y, .... In question (14) the answer [p. 456] that S = ij (x 2 + y' 2 -x 2 -y 2 )dt 
expressed as a function of x, y, a, b, t will satisfy simultaneously the proposed equations— 
would be, not of course a complete answer, but a good and creditable one; without 
the words “expressed as a function of x, y, a, b, t” it would be altogether worthless. 
A clear and precise indication of a process of solution is very much better than a 
detailed solution incorrectly worked out. 
C. VIII. 
58