DOUBLP] GAMMA FUNCTION. 
31;l 
and by the previous paragraph, the expression on the right-hand side of tliis relation 
will evidently reduce to the form where is some constant whose value may 
he readily seen to he given hy 
gj — hViCOy 
COi 
fl TTL {ni 
+ 
This value simplifies on detailed consideration. 
When il(wi)l > 
according as I(wi) is positive or negative. 
When lI(wo)| > there are four subsidiary possibilities ;— 
(a) When lies within the axes to — 1 and — 
(/3) ,, Wj ,, ,, ,, 1 and 
(y) ,, ,, — L and — 
and (S) ,, — (x)y ,, ,, — 1 and a»o. 
In cases (a) and (S) §1 — h'>?i = ± the upper or lower sign being taken 
according as I(w,j) is positive or negative ; and in cases (^) and (y) the upper and 
lower signs are interchanged. 
We thus see that in all cases 
^.•^1 ^ _ () ~ hh '^2 
SO that we have tlie required ecpiation 
cr (s + Wj) = — cr ('«;). 
Similarly we find cr (z -|- wo) = — a (z). 
The verification of these results affords substantial proof of the general correctness 
of the signs which are involved in the work. 
§ 36. It is interesting finally to notice that just as the gamma functions do not 
exist when r = coJoj^ is real and negative, so the elliptic functions demand that t 
shall not he real. 
The condition that r must not he real and negative arose explicitly at several 
stages, and might have been jjredicted a 
For, when is real and negative, it is obvious that 
n =z + moOJ., 
~ ~ U/r, z= 0, 1, 
will have a zero value at least once. 
And thus the function 
00 
oo 
^ I excluded. 
oJ 
00 00 
■n — 1 / \ 2 -i-r -f-T / 
To ^ {z) = . z . U n 
rill = 0 Wg = 0 
1 + jji« 
_ L+ ill 
fi 2n2 
will be infinite independently of 2 ; that is to say, it ceases to exist. 
VOL. cxcvr.— A. 2 s 
