[657 
M =1: we have 
658] 
143 
a x 2 ' 
■mation 
; then, writing 
of #* +1 = 0, 658> 
or, if a) be an 
= 0 ; so that a 
ON SOME FORMULAE IN ELLIPTIC INTEGRALS. 
v = 0 3 a, (1 + 2a>), 
(o 4 6 4 = — (o; and 
[From the Mathematische Annalen, t. XII. (1877), pp. 369—374.] 
I reproduce in a modified form an investigation contained in the memoir, 
Zolotareff, “ Sur la methode d’integration de M. Tchebychef,” Mathematische Annalen, 
t. v. (1872), pp. 560—580. 
Starting from the quartic 
(a, h, c, d, e)(x, l) 4 , = a.x — a.x — ß.x— y.x — 8, 
we derive from it the quartic 
ed. We have 
= V — 3, which 
(Uj, hi, Cj, di, Ci) (xi, l) 4 oq . Xi «i. x 1 ßi. Xi — yj. Xi — 8i, 
where, writing for shortness 
\ = — & + /3 + <y — 8, 
i relation cor- 
3 integral, and 
he modulus, a 
a — ß + y - 8, 
v = a + ß — y — 8, 
the roots of the new quartic are 
*“'+£■ 
^ = e, 
6 being arbitrary: the differences of the roots a lf ß 1} y 1} 8 l are, it will be observed, 
functions of the differences of the roots a, ß, y, 8.