208 
MR. E. AV. BARNES ON THE THEORY OF THE 
tlie succeeding investigations however liave, I Ijelieve, 1)een undertaken or suggested 
l^y other inatheinaticians. 
In tlie tirst pa})er"'" pnl)lished in the connection I attempted to give a homogeneous 
theory of tlie ordinary gamma function, considered from tlie point of view of 
WeiePuSTUASs’ function tlieory. I introduced a parameter w, and showed that the 
theory was suljordinate to that of a function satisfying the ditference equation 
f{z + to) — t\z) = ih 
s l)eing any conqdex quantity. 
That theory led naturally to the considerationt of the G function, satisfying the 
ditference etpiation 
G(.+ i) = r(.)G(^), 
and sul)stantially a function all of Avhose properties could l^e obtained by ditferentiating 
tlie simple gamma function witli respect to the parameter. 
1 next considered^ an extended function G( 2 /r) satisfying the two functional 
relations 
f(z + 1) = r(: ,/■(•.); f(z + r) = r(2) (-in) T--*V'C). 
and reducible to the G function when r = 1. Several points in that paper suggested 
the formation of a symmetiical double gamma function, in which r should be replaced 
by the quotient of two parameters Wj and Wo. In the present investigation such a 
huiction is defined, and its theory developed in, I hope, complete detail. The 
function is the natural extension to two parameters of the simple gamma function 
l’l(2 I «)■ 
It is necessary for a complete exposition of the theory to consider the properties of 
wliat I pro})Ose to call doidjle Bernoullian numl)ers and functions : functions which 
are the natural extension to two parameters of the simple Bernoullian functions, 
considered in Part II. C)f the earliest ])aper of the seiies. 8ucli a theory is develojoed 
in Part I. of the present paper. 
In Part II. I consider the elementary theory of the double gamma function. It 
is shown that certain symmetrical modidar constants arise as finite terms of 
asymptotic expansions in a majiner exactly analogous to the origin of Euler’s 
con.stant y. 
Such considerations lead naturally to Part III., in nhich are deduced from a 
contour integral, which is a doulde generalisation of Biemann’s 4 function, certain 
noteworthy asymptotic approximations, of which the most impoidant is an extension 
Barxes, “The Theory of the tTamma Function,” ‘Messenger of Mathematics,’ vol. 29, pp. 64-128. 
t Barnes, “The Theory of the Ct Function,” ‘ Quarterly Journal of Mathematics,’vol. 31, pp. 264-314. 
I Barnes, “Genesis of the Double Gamma Function,” ‘Proceedings of the London Mathematical 
Society,’ vol. 31, pp. 358-381. 
