842 
MR. E, BARNES ON THE THEORY OF THE 
When a is large and .s has any value, the left-hand side of this relation may be 
expanded in the form 
' rr'+i 
. +AM+ . . 
• ^,r-rs • ^ 
ce 
where the P’s are integral polynomials of s of degree indicated by their suffices. 
As they vanish for (^ + 1 ) values of s, we must iiave 
(n + w) — (a) — a ^ 
b^+i (g) , Px-+2 js) 
(f^s+k+l “t" + 2 
Thus t{s, a, oi) is convergent with S :- 
—, and is tlierefore convergent, 
provided 
|1 (5 + /^ + 1) > 1, or g (.s) > —A 
which is the condition with which we started. 
There is one point which does not arise in the work of Melltx, who takes the case 
oi = 1 . It is that throughout we must work with man}-valued functions with s as 
index, which have their principal values with respect to tlie axis of — w. For in 
expanding (a + w) — (a) — «“h where a is large, we have tacitly 
assumed that log (a -{- to) = log a -f- log (^1 + 5 which, unless w is real, is not the 
case when a is large and nearly real and negative, so that a and a + w lie on oppo¬ 
site sides of the axis of — 1 , if this axis is the axis of the logarithms. 
§ 57. It is now possible to construct tbe double Piemann { function by extending 
the previous analysis. The function so constructed might be made fundamental in 
the theory of double gamma functions and double Bernoullian numbers, these 
functions arising for particular values of the variable s. We will indicate the 
development of the theory from this point of view, for brevity establishing only the 
principal results, or those which, as in §§ 54, 55, have been established only over part 
of the a plane. 
The double Riemann ^ function ^ 0 ( 5 , a|(yj, Wo) is tbe simplest solution of tbe differ¬ 
ence equation 
f{a -f- ojj -f ( 0 .^ — f[a-\- w^) —f{ct + ( 0 . 2 ) -l-_/(o) = 
a, s, 0 )^ and wo having any complex values s\ich that wo/wi is not real and negative. 
T’ne determination of a~^ will appear in tlie course of the investigation. 
In the first place, it is evident that when g (s) > 2 , a solution is given by the 
series 
00 CO 1 
^ V __ 
Hij = 0 ,1)^ = 0 + m^CO^ -t- ’ 
which will then, by Eisenstein’s tlieorem, be convergent. 
