DOUBLE GAMMA FUNCTION. 
201 
T 1 
len 
7^1 (wj, M,) = 
n= T> L^l 
1 " 1 
— {lognwo 4- log?;.aii — logri((yi + wo)} — 2) ^ TT 
" m,=0 m,= oG" 
But this expressioji is symmetrical in co^ and Wo; and we must therefore liave the 
analoo;ons relation 
w 
here m = 0, unless 
-f CJ3) ~ — ~ "1“ 
co.^ does 
O/i TTi 
W, 
lie within the region bounded by the axes 
(<^i + con^ does not 
from the origin to — and —1, in which case ni = dr B the upper or lower sign 
being taken as I (w^) is positive or negative. 
Provided therefore that 
721 (*^0 ^-*2) — 
>; = 00 
1 n 1 1 
logn ~~ S %' ”3 d“ (") *^21 
WiCO.i 
COiW,, 
we have, with the assio-ned values of m and m , the two difference relations 
&)o 
The function (w,, we propose to call the first double gamma modular form. 
It will subse<[uently be expressed in terms of the function D(r) introfhiced into the 
theory of the functions G (2:|t) (“ Genesis of the Double Gamma Function,” § 4 ). 
It will be seen later that the alo'ebra of the double namma function would have 
O CO 
been slightly simplilied had a modified value been taken for this function 731(01, o,), 
and the analogue sliortly to be considered, 733(01, 03). I did not observe tins fact 
until the theory liad l)een completely devek)})ed, and the matter is scai'cely of suffi¬ 
cient importance to demand the labour whicli such a change would entail. 
CoroUarij. —Notice that it has been proved incidentally that 
n n 1 
V n;'' ___ 
«!i=o 
is infinite, when n is infinite, like --— loo- n. 
§ 22. As the numbers in m and m' enter constantly into the analysis, it is necessary 
to consider their pro])erties. 
Suppose that the functif»ns log 2;, log^y, log^y are natural logarithms (with e as 
base), which are real when 2 is real and positive, and which are rendered \miform by 
cross-cuts along the axes of — I, —-Oj and —03 respectively. 
2 P 2 
