340 
[769 
769. 
ON A FOEMULA EELATING TO ELLIPTIC INTEGEALS OF 
THE THIED KIND. 
[From the Proceedings of the London Mathematical Society, vol. xiii. (1882), 
pp. 175, 176. Presented May 11, 1882.] 
The formula for the differentiation of the integral of the third kind 
in regard to the parameter n, see my Elliptic Functions, Nos. 174 et seq., may be pre 
sented under a very elegant form, by writing therein 
sin 2 (/> = x = sn 2 u, sin cj) cos cf) A = y = sn u cn u dn u, 
and thus connecting the formula with the cubic curve 
y 1 = x (1 — x) (1 — №x). 
The parameter must, of course, be put under a corresponding form, say n = — —, 
CL 
where a = sn 2 6, b — sn 6 cn 6 dn 6, and therefore (a, b) are the coordinates of the point 
corresponding to the argument 6. The steps of the substitution may be effected 
without difficulty, but it will be convenient to give at once the final result and 
then verify it directly. The result is 
dd a — x du m — a 
We, in fact, have 
^— = 2 sn u en u dn u = 2y,