DOUBLE GAMMA FUNCTION. 
371 
j-ai 
j log Ug (a) da — log (co^, co^) + (m + m') 38/(0) — rtvlm 
S/(o I W. 2 ) 
_ A [ (-- Mlog(- + 7} 
Stt J 1 
X — 
dz. 
Since = y/, we see that, to establish the formula of § 73, it is necessary to 
show that 
!_ ( (- Mlog(- U + 7} 7 _ 1 Piico) , CO 
o I 1 r — CO loff-" -p “ 
STrJy^ 1 — e ^Pi((w) ' 12 
This may be readily done as follows 
We have 
log- 
f g '‘d— Mlog ( — ^) + 7} 
27r J 
po{co) 27r JL (1 — e 
00 
Integrate with respect to a between co and 2oj, and we find 
dz 
f i [ c 
log r 2 (a I w) c/a — ajlogpo(w) = — - 
J ^IT J 1 
i [ e z) 2{log(- z) + 7} 
1 — e 
dz 
{-z) -hiog(-g) + 7} 
llT . I 1 — 
dz , 
for it may be readily seen that ^ dz 
0 . 
[This vanishing contour integral occurs when Raabe’s formula is proved by means 
of the expression of the simple gamma function as a contour integral.] 
But as we have deduced in § 75, from Alexeiewsky’s theorem. 
h / I \ 7 1 , \ 1 Pdco) CO 
logT.{a\co)da- colog p.^ico) = colog-^^ + - 
The contour integral has, therefore, the required value. 
We here conclude our investigations of the algebra of the double gamma functions, 
It is evident that the formulae admit of still further development; they lead, for 
instance, to many curious relations between the integrals obtained in Part III. Such 
considerations are, however, foreign to our immediate purpose. The development of 
the integral formulae and the theories of multiplication and transformation in the case 
of the double Riemann (, function is interesting in that we thus combine many of the 
formulae which have been obtained separately for Bernoullian and gamma functions, 
and the algebra by which such developments are obtained by the extension of 
Mellin’s definition of the simple C function is in many ways attractive. Owing, 
however, to the length of this paper, we do not propose to consider it in this place. 
3 B 2 
