24] 
ON THE INVERSE ELLIPTIC FUNCTIONS. 
153 
whence 
/0 {</>- (x+a) - ft 2 (x - a)) dx = 
2(pa fa Fa 6~x 
1 + e 2 c 2 p 2 ap 2 x 
the first side of which is 
/-a 0“ + a) dx — f a (jy (x — a) dx — 2 f (l p 2 a da 
m 
(6i). 
Hence, multiplying by e 2 c 2 , and observing the value of (§x, 
(Si / (a; + a) ^ (&' (oc — a) (£x'a _ 2e 2 c 2 fa Fa (f>a p 2 x 
(& (x + a) (&(x —a) <3r a 1 + e 2 c 2 6 2 ap 2 x 
If in this case we interchange x, a and add, 
<B'x <$fa 
(Hxx (Sira. 
(S' (# + a) 
C5 (x + a) 
= e 2 c 2 pa px p(a + x) 
(62). 
(63). 
[By subtracting, we should have obtained an equation only differing from the above 
in the sign of a.] 
Integrating the last equation but one, with respect to a, 
(# + a) + ¿(¡S (x - a) - 2Ufitx — 2l(&a = l (1 + e 2 c 2 p 2 xp 2 a), 
the integral being taken from a = 0. Hence 
& (x + a) dBr (x — a) = (Hr 2 #© 2 « (1 + e 2 c 2 p 2 xp 2 a) (64) ; 
or 
whence also 
(Hr (¿c + a) CB (x — a) = (£\ 2 x(Fx 2 a + e 2 c 2 y 2 ^y 2 a, 
► 
7 (# + a) <y (x —a) = y 2 xdH 2 a — <y 2 a (& 2 x, 
g (x + a) g (x — a) = g*xi& 2 a — c 2 g 2 a (& 2 x, , 
G(x+ a) G{x — a) = G 2 xd5 2 a + e 2 G 2 a<&rx, 
(A 
these equations being obtained from the first by the change of x into x + - , x 4- ^, 
ö) VI 
x + 9 + - -. They form a most important group of formulae in the present theory. By 
integrating the same formulae with respect to x, and representing by n (x, a) the 
, f — e*c 2 6a fa Fa p 2 xdx T . . ,. . 
integral -.00.0 .0 , Jacobi obtains 
6 Jo l+e 2 c 2 p 2 ap 2 x 
n( „ a) i,ffi(g-a) . C'a. 
11 ■ ’ a) ~ 2i ffi(z+a) + ffia" 
an equation which conducts him almost immediately to the formulae for the addition 
of the argument or of the parameter in the function n. This, however, is not very 
C. 
20