489] 
AND THE (2, 2) CORRESPONDENCE OF POINTS ON A CONIC. 
17 
Now starting from the differential equation 
dx _ + dy 
V{(a, b, c, d, e\x, l) 4 } ~ V{0, k c, d, e\y> l) 4 } ’ 
the integral equation is known to be 
b, c, d, e\x, l) 4 } — b, c, d, e\y, l) 4 } 
x-y 
where 6 is the constant of integration. Writing, for 
Y= (a, b, c, d, e\y, l) 4 , this is 
= a (x + y) 2 + 46 (x + y) + 6 6, 
shortness, X = (a, b, c, d, e$jx, l) 4 , 
X + Y — 2 *J(XY) = a (x 2 — y 2 ) 2 +4b(x — y) (x 2 — y 2 ) + 6 8 (x — y) 2 ; 
or, what is the same thing, 
a (¿c 4 + 3/ 4 ) — 2 \/ (XY) = a(x 2 — y 2 ) 2 + 4b (x — y) (x 2 — y 2 ) + 66 (x — y) 2 , 
+ 46 (x 3 + y 3 ) 
+ 6c (x 2 + 3/ 2 ) 
+ 4# (» + 3/) 
+ 2a, 
viz. this gives 
V(IF)= aafy 2 
+ 26 (¿r 2 ?/ + ¿t'3/ 2 ) 
+ 3c (x' 2 + 3/ 2 ) 
+ 3# (# — 3/) 2 
+ 2d (x + y) 
+ e, 
and, rationalising, the integral equation becomes 
— 6adx 2 y 2 
— 4 adxy(x + y) 
— ae(x + y) 2 
+ 4 b 2 x 2 y 2 
+ 12bcxy (x + y) — 12bdxy (x + 3/) 
— 8bdxy 
— Ybe (x + y) 
+ 9c 2 (x + y) 2 — 18c# (x 2 + y 2 ) 
— 12c# (x + y) 
+ 9 6 2 (x — y) 2 — 12dd (x + y)— 6ed + 4# 2 = 0; 
C. VIII. 
3