DOUBLE OA^FMA FUXCTIOX. 
309 
And hence 
V — li-it 
,l = » 
_ V 
U ''U 
03 
/<?> — — i? i/L>=—i’ “ " 
log((yj -h w.,) + + w.,)] 
L — (<^1 — ^ 2 ) — [— (^1 — -I - 
Now, as may lie readily seen 1)y examining the different possilffe cases in a 
diagram, 
l0g(wi + OJo) ff- log [- (Wi + W^)] - log( wi - CO 2 ) - log[- (Wi - Olo)] 
1 &)i + W.I 
= 2 loo- - -, 
where that value of is to he taken whicli is on the same side of the real 
axis as 
With this proviso. 
V = 
_ V V' Jl 4_ 
_,C ft" CO^Q).-, 
“ l0£.'‘^+“’ 
coi a) .T 
the infinities in the donhle summation being equal in absolute magnitude. 
§ 32. We may now express Weterstrass’ ^ function in terms of derivatives of 
simple and double gamma functions. For on integrating the relation obtained in the 
previous paragraph we find, 
remembering that 
:^uz) =- p(A* 
rl 
— i{^) — i i "n i ^ 2 ) i cjj) — ± — - + r: + p, 
where p is constant with respect to s. 
Making then z = 0, we find 
0 = — Syo 2 (d:: Wj, dz oj,) — — (^og(wi) — y\ + “log(— w,) — yj 
- ~ {log Wo - y] + ;7 ■! — yI + g- 
(i).l Wo 
Now by § 23, 
-yio(± "i, ± — 9 
_ Wj 4 Wo [ log (wj 4" Wo) - log W^ - log Wo 
Wj^wo [ — log ["— (^i “k ^•’)J ”k (— ^ 1 ) ~k ( — Wo) 
&)| — Wo [ leg (wo — W[)-log (— Wj) — log Wo 
+ 
2wjWo [ — ^eg’ (w| ojo) “h ^eg ojj ^ ~)“ ^og' (Wo) 
= ( 7 ^ -f -y)[log(wi + Wo) — ]og[—(w^ + Wo)]! 
:w, 
ZWo 
4-{leg(— w,) — logw^ j 4“ ^ 0 ) 
w, 
1 
-IW, ZWo 
~ ^h) — l'^g(wi — Wo)j. 
* Jordan, ‘ Cours d’Analyse,’ 2ii(l edition, p. .347. Xote that .JoRD.tx uses 2wi and 2cu.2 instead of 
ui and 0)0 of this paper. 
