370 
ME. E. W. BAENES ON THE THEOEY OF THE 
Since, by § 64, 
% P -2 ("n (^ 2 ) — 273-t 28 /( 0 ) = i log 717^73 + i [1 — 2 Si'(f>)] log 2 , 
we see that this formula may equally be written 
I log r.i {z "h Cl I cl)]^, ojg) clz 
— — a log 
r,(« I Wo) 
Pi ("2) 
+ I log-7172/3 + m [1 -28/(0)] log 2 
0^ 
*> 
O 
“h (1 “h 2'nnn) j Sj (a | oio) “8 f — 
log 
ro(« + Wo 1 Wo) 
P2("2) 
This and the corresponding formula, obtained by the interchange of w^ and lo.j, 
m and m , are tlie analogues for double gamma functions of IIaabe’s formula for 
simple gamma functions. 
§ 77. In the particular case when a is positive with res 2 :)ect to the w’s, it is possible 
to obtain more sinqjly the value of 
r<oi 
log To (a I W|, Wo) da 
by means of the contour integrals investigated in Part III. 
We give tliis method of proof as it leads incidentally to an exj^ression as a contour 
■ 4 . 1 r 1 ^2 
mtep’ral tor Iop’ —;—-. 
We find, on integrating the exjDression for log —- given in § 45, 
pn (wj, Wo) 
“1 
•' 0 
log To («) da — log p.^ (wj, 0 J. 2 ) 
= (M + m + ni) 2771 [ oS^ («) da + .•>8/(0) 2 M 7 rt — [ 
J 0 ~ ~ 277 J 
^ r (-s) M‘iog(- z)+ 7 } 
IL (1 — 
and the right-hand side, by an ai^jilication of the formulae of §§ 6 and 44, becomes 
S/(0 I Wo) 
dz, 
(M + 7 n -}- 711') 2771 
— faj| 08/ (0) -p 
-\r 08 /( 0 ) 2M77t 
_ I- r (— wMiog (~ 27. 1 -f- 7 } 
27^] 1 
(1 - 
dz 
Now, reducing the contour to a small circle round the origin, we see that 
' (M + m') 2771 f I «2) 
Ztt ' 
and therefore 
, 1 — c “ 
W2 
(M + 771') 2771 , 
