32(5 
MR. E. W. BARNES ON THE THEORY OF THE 
Now, by considering the various cases which can arise, it may he readily seen that 
M d- r,i + = 0. 
These constants all vanish unless and embrace the axis of — 1. When 
this take place, suppose that lies above, and Wo below, the real axis. 
Then 
/ 
[X — 
m' =- 
> 
M = 
iJ 
f 
jX — 
o-> 
1 
m' — 
0 
M = 
oJ 
when (oj^ -p 0 ).^ 
when (wj + oj.,) 
lies above and 1/L below the real axis. 
and l/L both lie below the real axis. 
/=-ll 
m = 1 >when (^1 + and 1/-L both he above the real axis. 
M = 0 J 
on — 0 
M= 1 
j>when (oj^ + (Do) lies below and 1/L above the real axis. 
We get similar sets of values when the imaginary parts of and have opposite 
signs to those just assumed. 
In all cases 
M + m -j- = 0, 
and, therefore, with the values assumed for X, and X,, 
Eg ^ + <^]) _ rpc I I Mj) 
To ^ p\ (^2^ 
Similarly we should find 
Fj ^ {a 
But these are identically the fundamental formulge for tlie double gamma function 
found in § 23. 
The values assumed for \ and Xo are therefore correct. 
We have, then, the two important formulae 
■f ro(c. I (0-1, ft).,) 
log ^^ 
P2 
oSo («') (M + m + in') '2 ttl -p oS/ (o) 2M77t 
I _L f 1-WMlog(-«) + 7 } 
27rjL 
(1 — (1 ~ 
dz, 
and log po (ct)j, 0 J 2 ) = — 2M7rt oS'i (0 | a»o) 
Li 
= 0 
C-"" (— 2) Mlog (- ~) + 7} 7 . 1 
, -^^1-A (/2 -p log a 
L (1 — (1 “ ® 
