ON THE INVERSE ELLIPTIC FUNCTIONS. 
154 
[24 
closely connected with the present subject. For some formulae also deduced from (63), 
i — a) (&(y — a)<&(x + y + a) . . 
by which ^sr-7 ^ ^ f is expressed in terms ol the function <6, see 
J (x + a) CS (y + a) Crr {x + y - a) Y Y ’ 
Jacobi. 
Note.—We have 
y ß x = 2:1111 ( 1 + 
(m, n) 
g ß x = IUI 11 + j= 
(ra, n)J ’ 
the limits of n being + q, and those of m being + p, in the first case, and p, —p — 1, 
in the second case. Also - = oo . 
We deduce immediately 
( 
7/s \ æ + ö = ®+ô nn j 1 + 
x + 
{m, n) 
= nn ! 1 + 
(m 
X \ ' ü 
; Tl)) ' î 
co (m, n) 
2 (m, n) 
(paying attention to the omission of (m = 0, n = 0) in y$x, and supposing that this 
value enters into the numerator of the expression just obtained, but not into its 
denominator). This is of the form 
7 ,(, + |) = ^nn(i + ( ^) ; 
but the limits are not the same in this product and in g$x. In the latter m assumes 
the value —p — 1, which it does not in the former; hence 
7ß « +9ßX = A-r nj! + 
— (p + co + nvij ’ 
and the above product reduces itself to unity in consequence of all the values 
assumed by n being indefinitely small compared with the quantity (p + ^); we have 
therefore 
7/3 
(*+!)=¿a, 
•(65). 
and similar expressions for the remaining functions. To illustrate this further, suppose 
we had been considering, instead of y^x, the function y-px, given by the same formula, 
but with ^ = 0, instead of - = oo. We have in this case also 
q q 
7-/3 [ æ + s ) <7-/8 x — A' -f- n w 1 + 
( — P + i) w + wW ’ 
A' different from A on account of the different limits. The divisor of the second side 
takes the form 
{æ-(])+|)û)}Il(l + 
x-(p + $)co\ . 
nin 
+(-*>+*)«■ n (i +, 
V nvi 1