144 
ON SOME FORMULÆ IN ELLIPTIC INTEGRALS. 
[658 
We assume a a =a = l, nevertheless retaining in the formulae a x or a (each mean 
ing 1), whenever, for the sake of homogeneity, it is convenient to do so. The relations 
between the remaining coefficients b x , c 1( d x , e 1} and b, c, d, e, are of course to be 
calculated from the formulae —4b = %a, 6c = 2a/3, &c., and the like formulae — 4b x = 2»!, 
6c l = %oi 1 /3 1> &c. We thus have 
-ib, = 4,e +i Sp 
6c, = 6# + f0 2^ + iSX 2 , 
A. 
- 4d x = 40 s + f 6- 2 ^ + J02A 2 + %\/xv, 
A 
Ci = 0*+ |-0 3 2 + £0 2 2a 2 + 
A 
where 2 ~ = —- 2X 2 //, 2 . 
A A/ii> 
Writing, for shortness, 
C = ac — 6 2 , 
D = ctfd — 3a6c + 26 3 , 
E = a 3 e — 4 a 2 5c£ + 6a6 2 c — 36 4 = a 2 / — 3(7 2 , 
I =ae — 4>bd + 3c 2 , 
J = ace — ad 2 — 6 2 c + 2&a£ — c 3 , 
„ — a 2 / 4- 12(7 2 
5 = 
we have 
2A = — 4 (& 4- 8), 
2A 2 = - 48(7, 
2A/i = 24(7 + 8 (b + By, 
\/xv = 32D, 
2A> 2 = 64 (— a 2 / 4- 12(7 2 ), 
where the last equation may be verified by means of the formula 
(2X/t) 2 = 2A 2 // 2 + 2A/6i/ 2A. 
And we hence obtain 
«i= 1, 
b x =-e - b, 
c x = 0 2 + 2Be - 2(7, 
d 1 = -6 s -SB6 3 + 6(70 -D, 
* = 6* 4 4£0 3 - 12(70 2 + 4Z>0.