DOUBLE GAMMA FUNCTION. 
361 
This may be regarded as the transformation formnla foi' the doulde Stirlin 
function (cu^, w^). 
Notice that when p^q — m, = 0 , oj.y), and there¬ 
fore the preceding formula liecomes 
vi m 
, //-wj -t- sw, 
log n n r. 2 ' — 
log p., (oj^, w.,) — log pA~, -- 
=0 s=0 ' / o r~ \ i ./ or- 
"h (l — "//U) 27rt -}- 7/i) oBj^ (^n ^ 3 )* 
Comparing this with the result obtained in § 63, we find 
J) = log di (^ 1 ^ + [1 - .S/ ( 0 )] log m. 
This result may also be obtained by the transformation of the line integral 
which expresses log (wj, Wo). 
§ 69. We have still to consider the transformation of the first and second double 
gamma modular forms 
(<^ 1 , Wo) and yoo (wj, w^). 
With this object we write symbolically 
X X 
//Zj = 0 T<?.> = 0 
/ (t), . ct).,V" 
///^—‘ + //r,—- 
V P ~ >1 ' 
and we write 
-f where m > 1 
r=0 s = 0 V P 
so that is a form analogous to the modular forms introduced into the theory of 
elliptic functions by Abel. 
By § 29 we know that within a circle of sufficiently small radius there exists the 
expansion 
log bs A) = - log ^ - '^720 - y yoi - S S' , -f S S'Jo, - . . . 
Take now the formula of § 67 
, ^ / I ft), (t)o\ ■'U' W 1 / I '''’Wi , s&).i\ 1 /^’^i 1 
log To 2 = S S log To 2 + —^ H-^ - S S' log r, —^ + - - 
^ \P qj r = 0s = 0 ^ ~\ p <lJ r = 0s = 0 ^ '\P 7, 
2iTTipp,ii 
a)o 
P ' U 
and expand in powers of z. 
VOL. CXCVI.-A. 
3 A 
