769] ON A FORMULA RELATING TO ELLIPTIC INTEGRALS OF THE THIRD KIND. 341 
and thence 
that is, 
Also 
and hence 
dy 
du 
dx 
Vch = 2 ' J ’ 
y ^ = 2® [1 — (1 + k 2 ) x + k-x-]. 
cn 2 u dn 2 u — sn 2 u dn 2 u — k 2 sn 2 u cn 2 u 
= 1 — 2 (1 + k 2 ) x + dk 2 x 2 , 
- y - 1 
rJ/ii m rt (n n\2 ] ' ' rl'i/ U rJ')! 
du x — a (a — xf ( du J du) 
= -——- {- x — a + 2 (1 + k 2 ) ax + k 2 x z — 3k 2 ax 2 ). 
(a—xf 1 
Interchanging the letters, we have 
A —A- — 7 —{— x — a + 2 (1 + k 2 ) ax + k 2 a 3 — 3k 2 a 2 x), 
dd ci —x (a — x) 2 
and hence, subtracting, 
A _A A V— = 7 1 — \k 2 a? - 3k 2 a 2 x + 3k 2 ax 2 - k 2 x 3 } 
eld a- x dux- a (a-x) 2 
(a — x)‘ 
— k 2 (a — x), 
k 2 (a — x) 3 
which is the required result.