500 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
[537 
and thence 
if 
wherefore 
Moreover 
2x = K V(1 + y~) — Ly, 
K = Q W(i + n ~) + n ) - q {V(i + ft 2 ) — n }> 
L — Q {V(l + ft 2 ) + ft) + q {\/(l + ft 2 ) — ft}j 
L 2 -K 2 = 4. 
that is, 
or, what is the same thing, 
(2x + Ly) 2 = K' 2 (1 -t- y 2 ), 
4a: 2 + (L 2 — K 2 ) y 2 -P 4Lxy = K 2 , 
x 2 + y 2 + Lxy = I (L 2 - 4), 
which is the rationalised form of 
q _ {x + V(1 + Q'~) 1 [y + V(1 4- y~)) 
n + V(1 + ft 2 ) 
And if Q=1 then L = 2 \/(l +n 2 ), \ (L 2 - 4) = ?i 2 , so that this equation is 
x 2 + y 2 + 2xy V(1 + ft 2 ) — n 2 = 0 ; 
or, when (7=0, the complete integral is satisfied by 
{x+ y(l+Q} {y + V(i + y 2 )} = ] 
n + V(1 + ft 2 ) 
that is, by 
x 2 + y 2 + 2xy v'(l + ft 2 ) — ft 2 = 0. 
We may without difficulty rationalise, and present the result as follows: the equation 
2 (*+!) + („_ I),3, 
1 + — )dx 
x 2 
+ i 2 ( y + }) + ( M - s) (* - s)} i 1 + ? 1 dy = °- 
has the complete integral 
2/ I I n — -Ì - in 2 — = G + 4 log — , 
« y- \ x) \y y) \ n) \ n 2 ) ° n 
and a particular integral xy — n- 0 : the complete integral is in fact 
(71 - a?y) {- n s x 2 y 2 + n 2 xy (- ocr — y 2 + 1 ) + 71 (a; 2 // 2 - a: 2 - y 2 ) - a;y} = ar 2 y 2 ?i 2 (C+4< log 
satisfied, for (7 = 0, by xy — n — 0.]