324 
WTJ. E. 4Y. BARNES ON THE THEORY OF THE 
and it is obvious that we may rejdace log (— 2 ;) in the subject of integration by 
log (— 2 ) + y without altering the value of the integral. Integrate successively 
witli respect to a, and we obtain 
log ro(«laJi, w.) = 
(1 — C 
dz + ( 1 , 
where tlie coefficients of the additive quadratic form are constants with respect to a. 
Itemember that 
lA [olb (rt j w.)] = 1 , 
then we evidently have 
(1 = 0 
log r,{a I Wj, CO.) = 
■Lt 
a = 0 
(1—c “1^(1 — 
L f «-«-■(- 2)'l{l0g (- Z)+ 7 } 
-I 
27r J 
-f- log a 
IL (1 —e “^b 
We now define tlie third double gamma modular form co.) by the relation 
log W;:) = — 2M7n oS^j(o | co^, co.) 
- Lt 
a = 0 
L f c 2 ) Ml»g(- -) + 7} 
dz "h log a 
27r J L (1 ~ e “ib(l — c;~“-b 
and we proceed to show that the constants and X. are such that 
=oSo(a) (M + m + m'yiTTi + oS'i(o)2M77t 
I Wf WMffig(-^)+7} 7 . 
’^27rjL (l-(?-“ib(l-c-“^'--) 
, r„(cb copco.) 
log --- 
” Pd/Oi, CO.) 
where the numhers m and m have tlie values assigned in Part II. 
If this relation is true we shall have 
1 I o fie/ 4 - CO] ) 
loff 
— S'] (a 1 CO.) (M + m + m')27n + 
C r C «'■( - [log (-2) +7} 
1—C“ 
dz 
§ 44. Let us now consider this integral. 
It is to Ije taken along a contour embracing the axis L, which we take to be the 
bisector of tlie smaller aiinle lietween the axes of ~ and —, unless such bisector 
^ CO] CO, 
should be the axis of — 1, in which case we take it to be nearl}^ coincident with this 
line. And in tlie suliject of integration that value of log (— 2 ) is to be taken winch 
is real when 2 is real and negative, and is limited by a cross-cut along the axis of L. 
Let us consider the relation of this integral to the integral 
^ I- ],-«q_,:)-l|P]g(_ ~)+ 7 } 
27r]j^ (1-c-^^b 
which is defined in the same way with reference to the axis of 1 /cjj.. 
