148 
ON SOME FORMULAE IN ELLIPTIC INTEGRALS. 
[658 
or writing y — a . /3 — 8 = y x — ci 1 . /3 1 — 8 U reducing and inverting, we have 
— F 1 = B — OL. X — 8 — \/X — ¡3 . X — r y] 2 , 
x 1 -8 1 ßi-8 1 .y 1 -ö 1 c n ’ 
which may also be written in the equivalent forms 
«i - & 1 
Xi _ gj y 1 - . Ä1 _ 
- 7i _ 1 
{V« — (3.x — 8 — \/x — y. x — a} 2 , 
{^/x — y .x— 8 — \fx — a. x — /3] 2 . 
~§i ~ Si. A - 
In fact, from the first equation we have 
*>-*>-■ = (ft-8.)(7.-8.) - K«-a.*-S- 
where the expression on the right-hand side is 
Si 2 - Si («i + A + 7x) + etiSi + & 7l — 2* 2 + #(« + /3 + y + 8) — aS — /3y + 2 y/X, 
X having here the value 
X=x — ol.x — ¡3.x—y.x —8. 
Writing for a moment 
P = a.8 + /87, Pi = ^8, + /3^, 
Q=/38 + ya, Q 1 = /3 1 8 1 + y 1 a u 
R = y8+ a/3, Ri = yi8 1 + a l fi 1 , 
then, by what precedes, Qi— R ly R 1 — P 1 , P t - Q x are equal to Q — R, R — P, P-Q 
respectively; that is, P x — P = Q 1 — Q = R x — R, = (suppose) 12, a function symmetrical in 
regard to 0(1, A, 7i5 a, /3, 7: the equation therefore is 
———p—-—— = Sj (Si — «j — /3 2 — 7j) — 2x- + x (a + /3 + 7 + 8) -1- 2 Vx + f2, 
X\ Oj 
or the relation is symmetrical in regard to a 1} /3 X , 7l ; a, /3, y: and the first form 
implies therefore each of the other two forms. 
Cambridge, 8 May, 1877.