368 MR. E. W. BARNES ON THE THEORY OF THE 
since the coefficient of log 2 is equal to 
— ^ ~ 17 + q" "h 1 “h •:)'2^/(^ I "35 ^3)+ I OJo) = 0. 
wCt}o —t -- 
We have, then, finally, 
[ log V^{z)dz = Wi[log 0 )^) — {rn + ni') 27rt 28 /( 0 )] + oj., log 
Jo Pl\^2l 
+ 2^/(1 + 2 m 7 n ). 
We thus see that substantially the doulfie Stirling function of and is 
expressed by — log Fg ( 2 ) dz, or, by symmetry, by - log Fo ( 2 ) dz. We have, in 
fact, the relation 
po(w;, w./ — {m + 7/f)27n oS/(o) = — [ log V.\z)dz — ^ log A 
Wj Jo " &)i 
l2co, “ 2?-n7n], 
where A is the Glaisher-Kinkelin constant (“ Theory of the G Function,” § 3) ; and 
therefore 
po(6Ji, W 2 ) — {m + m) 2771 oS/ ( 0 ) 
= - o f log Fo ( 2 ) dz - „ I log Fo ( 2 ) dz + —w log ; 
wy — &)o Jo <^1 —coy Jo J--Jvcoy — &>o ) coj 
(Wo 
lor, by § 22, log Wo — log Wj — 2(??i — m') ttl = log 
u/] 
§ 74. We may readily prove these results by the relation which exists between A 
and po {oj). [The latter is a convenient way of writing po(w, cu).] 
For, when each of the parameters is etpial to co, we have (“ Theory of the 
G Function,” § 29) 
r^^{z\co) = G{p{'27T)-Lod^ 
{Z-UlY 
+ i 
G(i) = F.r‘ I 1 CO ) (277)^07-^, 
and therefore 
Now from the multiplication theorem for the double gamma functions, when m 
we have 
= 9 
27r^" 
CO 
= pHo2) 2-A 
( 277 )i 
/ \ r\ 7 
Hence 
