(25) 
Hence we may put 
Z (e; (ON 0) meen Z (2) en Pl (ma | mo) P 
and we have thereby arrived at a one-valued function Z(z), as yet 
merely existing in that part of the z-plane where « > 2. 
It will be at once noticed, that the agregate of values through 
which the thus defined function Z(e) can be made to pass, essen- 
tially depends upon the convention made concerning the amplitude 
of mw + m'o'. It is only when < acquires positive integer values 
> 2, that this convention becomes wholly immaterial. 
The question now arises whether the function Z(z) can be con- 
tinued across the limit of the domain in which it is originally defined 
by means of the double series. This question may be answered in 
the affirmative, indeed, it: will be found that by converting the 
double series into a definite integral the required continuation readily 
presents itself. 
Let 2@ and 2m’ denote a pair of primitive periods of an elliptic 
function pu and let us put 
(x) 7 1 
wiu) == pu — OE Ao 
4." ru 
sin” 
@ 
Then, we consider the integral 
1 
i= qi pe) du 
22 uz! 
aud take it along a loop Z beginning and ending at w = oo, going posi- 
tively round the point w == 0 and enclosing the points ©, 2w,30,... 
its double linear part being drawn as closely as possible along the 
right line 0, @, 2o ... Under the restriction that the real part of 
z > 2 it follows from the application of Caucny’s theorem, that the 
integral Z is simply equal to the negative sum of the residues corres- 
ponding to the poles 2 of w (u). For these poles constitute the 
system of singular points that the subject of integration possesses in 
the region outside the loop. 
Thus, as we have in general 
P=2moa+2m'o' , 
ete eo 0, PY AE 2 
Em = ¥; ae age 
3* 
