537] 
that is, 
SOLUTIONS OF A SMITHS PRIZE PAPER FOR 1871. 
whence 
that is, 
Q" {2A- +1 + 2n V(1 + ft 2 )} - i {2ft 2 + 1-2» V(1 + A 2 )} 
= 4 {a? V(1 + «0 + y V(1 + 2/0} 
+ 4xy Q {V(l + ft 2 ) + n] - ^ {V(l + ft 2 ) - n) , 
4 [x V(1 + a: 2 ) + y V(1 + 2/0 + 2nxy — n V(1 + ft 2 )} 
= Q 2 {2ft 2 + 1 + 2» V(1 + a 2 )} - ~ {2ft 2 + 1 - 2n V(1 + ft 2 )} 
+ 8ftau/ — 4?i V(1 + ft 2 ) 
— 4^2/ 
Q {V(i + a 2 ) + w}-^{V( 1 + a 2 ) — ft} 
— (Q 2 — Qhj (^a 2 + 1) + (Q 2 + — 2j 2n V(1 + ft 2 ) 
- (Q - g) ^y V(1 + ft 2 ) - 4 (q + - - 2) nxy 
= (Q-1) (2ft 2 + 1) 
+ +1)2 2n V(1 + A 2 ) 
- 4 % + ^ xy Vd + A 2 ) 
(Q _ 1> ) 
- 4 — ^ ' ftaft/j 
= (Q — 1) 12 suppose, 
# V(1 + a; 2 ) + y V(1 + y 2 ) + 2fta;?/ — ft V( 1 + ft 2 ) = i (Q - 1) 12. 
And the integral equation is 
i(g-i)a + iog<2 = c, 
which, for (7=0, is satisfied by Q = 1. 
Now starting from 
{a; + V(1 + a; 2 )} [y + V(1 + f)} 
v ft + V(1 + ft 2 ) 
we have 
V(1 + a; 2 ) + X = Q |V(1 + ft 2 ) + ft} {V(l + y 2 ) ~ 2/}> 
V(1 + x 2 ) - x = g {V(l + ft 2 ) - ft} {V(l + 2/0 + 2/}>