44 
AN ILLUSTRATION OF THE THEORY OF THE ^-FUNCTIONS. 
[716 
Hence, writing y = z* = sn 2 u, we have 
2u = \J{d — b.c — a) I 
J a 
dx 
and it is to be further noticed that to 
x — a, b, c, d, 
correspond 
or we may say 
Writing for shortness 
we have 
and moreover 
sn u = 0, 1, oo , 
u= 0, K, K+iK', 2K + iK\ 
2 
V(rf — b.c — a) ’ 
cm 
dx 
aK = 
or if for a moment we write 
then these equations are 
a(K + Œ')=f 
J < 
a (2K + ¿JT) = f 
J t 
ra dx 
VW’ 
6 
• vw* 
dx 
VW’ 
V№’ 
= -4, &c., 
0 V(^) 
aK = B — A, 
ct(K + iK') = C-A, 
a (2K + iK') = D-A. 
Hence B + 0 —2A = D — A, that is, A — B — C + D = 0, or B — A = D — C, that is, 
b dx [ d dx 
f b dx _ f 
JaVCXj-Jt 
V(X) J c V(X)’ 
where observe as before that x = a to x = b, or x = c to x = d, X is positive, and the 
radical \J{X) is taken to be positive. 
We have also 
aK = B-A = 
aiK’ = C — B = 
dx 
dx 
b V№’