302 
MR. E. W. BARNES ON THE THEORY OF THE 
Now G 
CO, 
T ) satisfies the equation* 
f{z + Wj) _ i /.^ N 
= T «i - (Jtt) 
(O, 
r - / G) 
where r = e ; and the principal value of log t is to he taken. 
(The same remark, of course, applies to every many-valued expression of this nature 
which occurs in the course of the investigation.) 
Employing the relation (1) in conjunction with this equation and the equation 
we obtain 
Fq + (On) _ G(a;| Mj) Sife I u,) 
\/(27r/ft)o) 
Ri (2 1 Wj) " 
0-0-' ■ ' 
r-l 
= C"™" '(2^) ^ Or(2l(y^)a)G "IT (2'^) ’’ 5 
which i-educes to 
— TTL s ,'(2 I Wi) __ p 2?-?r!T + sys | w,) [log wj — log aij - log t] 
Eut we have seen (§ 22) that 
log (0.2 — log oj^ — log T = 2 {m — m') ttl 
for m — m' = 0, unless the difterence of the amplitudes of and a»o is greater than tt, 
in wliich case m — m' = i 1 , according as — I j is positive or negative. 
We thus find r = 0, and incidentally we obtain a valuable verification of our 
results. 
And now finally 
lN-i(2) = w.,(277)-sG(a;^e-'“’^‘)'^®“(^> g(- 
\"i 
the relation between the two forms of double gamma functions. 
§ 27 . From the relation just found we may at once express the gamma modular 
constants C(t) and D(t) of the former theory in terms of ( o .,) and yo]L(w|, Wo) 
respectively. 
For we have 
T 
G 
\«i 
r I =: c “"2 
\(> n 2 n~ 
where 
^ = 3 log 275 -t -f ^ log t + yr — C(t), 
TFT- 
= — T log T — 
^ b 
and also 
00 00 
T 2 ~\z) = _ j-j 
)Hj=0 7n.,=0 
1 + _)e n + 
* “ Genesis of the Double Gamma Functions,” §10. 
