508 
[88 
ON THE TRANSFORMATION OF AN ELLIPTIC INTEGRAL. 
[From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 204—206.] 
The following is a demonstration of a formula proved incidentally by Mr Boole 
(Journal, vol. II. [1847] p. 7), in a paper “On the Attraction of a Solid of Revolution 
on an External Point.” 
Let 
U = 
dx 
V[(l — x 2 ) [1 — (mx + ii) 2 }] ’ 
then, assuming 
ix = 
a + iy 
1 — iay ’ 
(so that x = ± 1 gives y = ± 1), we obtain 
1 
(l + a 2 ) (1 — y 2 ) 
(l-zayy ’ 
mx + n = 
(n — ima.) + (m — ina) y 
1 — iay 
Assume therefore 
whence 
— la ■■ 
ia + (n — ima) (m — ina) = 0, 
(1 — m 2 — n 1 ) + A 
2 mn 
(A 2 = 1 + w 4 + n 4 — 2m 2 — 2 n 2 — 2m 2 n 2 ),