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.
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