510 
ON THE TRANSFORMATION OF AN ELLIPTIC INTEGRAL. 
[88 
as may be immediately verified ; hence, assuming 
\z //1 — z 2 
y 
s/k\f \1 — X 2 z- 
vvhich gives 
we find 
that is 
Writing here 
then 
or if 
then finally 
dy 
dz 
U = 
0 V{(1 - y 2 ) (1 - khf)} s/k], V{(1 - z 2 ) (1 - \ 2 z 2 )} ’ 
dz 
V{(1 + — w 2 
dx 
(1-^)1- 
4 to 
(1 + mf — n 2 
dz 
-1 V[( 1 - OC 2 ) {1 - (mæ + n)} 2 ] J 0 V(1 - z 2 ) [{(1 + m) 2 - n 2 } - 4mz 2 ] ' 
X — COS 6, 2 = COS ^(f), 
dd 
W-/. 
d(f> 
o \/{ 1 — (m cos 6 + zi) 2 } J 0 \/(l + to 2 — n 2 — 2m cos </>) ’ 
r IZ 
m = - , n = , 
a a 
de 
dcf) 
sf[a 2 + {z + ir cos d) 2 } J 0 \J(a 2 + r 2 + z 2 — 2ar cos (/>) ’ 
the formula in question.