146 
ON SOME FORMULAE IN ELLIPTIC INTEGRALS. 
[658 
We consider now the differential expression 
this into the elliptic form, assume 
dx 
\/x — a .x — ¡3.x — y .x — 8 
; to transform 
k 1 2 = — a f, sn 2 a = 7 —^; 
y — a. ¡3-8 7 — 6 
{where a is of course not the coefficient, = 1, heretofore represented by that letter: 
as a Avill only occur under the functional signs sn, cn, dn, there is no risk of ambiguity). 
And then further 
a sn 2 u — 8 sn 2 a 
Forming the equations 
k 2 sn 
we deduce without difficulty 
sn 2 u — sn 2 a 
k 2 sn 2 a = — ^—> , k 2 sn 4 a = — 7 a ' a ^ 
p — 8 
sn 2 a — 
cn- a — 
dn 2 a — 
y-B./3-S* 
7 — a. sn 2 u _ x — 8 
7 — 8 ’ sn 2 a x — a. ’ 
a — 8 cn 2 u x — 7 
7 — 8 ’ cn 2 a x — a ’ 
a — 8 dn 2 u _ x — ¡3 
/3—8 ’ dn 2 a x — a ’ 
1 _ u sn 4 a _ (« — S) (- « + /3 + 7 -8) _ \(a-8) 
/3-8.y-8 ~ ¡3-8.y-8’ 
the use of which last equation will presently appear. 
We hence obtain 
2 sn u cn u dn u da = — (a — 8) sn 2 a 
dx 
(x — a) 2 ’ 
snMcnit dn u = sn a cn a dn 
\]x — a.x — 0.x — y. x — 8 
and consequently 
(x — a) 2 
2du = 
(a — 8)sna 
dx 
or, reducing the coefficient, 
cn a, dn a \/x— a.x — ¡3.x — y. x — 8 
dx 
\/x —a.x — /3.x —y.x —8 V 7 — a . /3 — 8 
du, 
which is the required formula. 
We next have 
2 0 _ 4 sn 2 a cn 2 a dn 2 a 4/3 — 8.7 — a y 1 — a 
sn ZiQj — 7— ; = = i- 1 , 
(1 — k 2 sn 4 a) 2 \ 2 7i — S/