428 
ON THE DOUBLE ^-FUNCTIONS. 
[697 
which equation, regarding therein A, B, C as determined by the three equations 
Am? + Bx + C = \JX, 
Ay 2 + By +C = *JY, 
Aw 2 + .Bw + G = V TF, 
is a form of Abel’s integral, giving s rationally in terms of x, y, w. 
Supposing that, when x = a, z — y: then the last-mentioned integral gives 
„ G 2 — abed 
a V* w = a?- 1 > 
where A, C are now determined by the equations 
Aa 2 + Ba +(7=0, 
Ay 2 + By + C=\JY, 
Aw 2 + Biu + (7 = V IF, 
and, imagining these values actually substituted, it is to be shown that the equation 
C' 2 — abed 
ay*w = 
A 2 - 1 
is satisfied by the value w = a. 
We have 
A.a — y.a — w.w — y— (a — w)\JY— (a — y) V IF, 
. a — y. a — w. w — y = (a — w) (a + w) VF — (a — y) (a + y) VTF, 
C.a — y.a — w.w — y = (a — w)aw V F — (a — y) ay V TF, 
or writing as before 
and also 
then F = a 1 b 1 c 1 d 1 , 
«-y, 0-2/, c-y, d-y = a l5 b 2 , c 1? d 1? 
a—w, b — w, c — w, d — w = SL 3 , b 3 , c 3 , d 3 , 
TF = a 3 b 3 c 3 d 3 , and the formulae become 
A = 
B = 
C = 
1 
(w - y) Viqä 3 
1 
{w - y) Va^ 
1 
(w-y) Va^s 
{VagbjcAi - Vad+Cod,}, 
{- {a + w) VasbjCidj + (a + y) Va^c^}, 
{ate VaabjCjdj — ay Va^c-jd^.