508 
MR. E. W. BARNES ON THE THEORY OF THE 
§ 31 . We proceed now to express Wetrk«trass’ elliptic functions in terms of clonble 
mamma functions. 
In Weterstrass’ notation of elliptic functions we have 
+ O)-’ 
5 
where Cl = + and and are the periods of 9 {z). 
Now by definition 
CO 00 1 
— 2 S S-. 
„)| = 0 ,,)2 = 0 (-• + ) ' 
llepresenting the various terms hy tlie corners of the parallelograms of the figure, 
we readily see that 
0 ).) = 
(z I Wj, Ct}.^ + [z I — <y^, {z \ — (x).^ + {z \ — co^, — at..) 
+ 
_ _oo (s -f o)" -X (~ + (:■ + 
3 + 
+ 
.3 
Hence, using the natural summation '%x}j.x^\z\ fi; i w.,) to express the left-hand 
side of this relation, v^e have 
{z \ ± Wi, ± Wo) = 9 '{z) -h Sipi^^^{z I ± Wj) -f- {z | ± Wo) + 5, , 
and therefore, on integration, 
9 {z) = Si//o® {z 1 ± ojp i Wo) — {z I ± Wj) — {z I ± Wo) + v + V, 
where v is constant with respect to 
Evidently 
^ = %n (± "n db - ETToS + '2 S 
1 
N ow 
r 1 
n n' 
i 
7:11 ("i> ^^^1 + ("i -1- "o)] — - - p p 
,/ = = 0 1R2=0v^ 1^1 ' //taCOj/ J 
whei'e the })rlnci})al values of the logarithms are to be taken. 
