266 Mr. E. W. Barnes. 



investigated in Part II of the memoir, two forms, 721(^1, ojo) and 

 722(^1, C02), are considered which arise substantially as hnite terms in 

 the approximations for 



n n -I n n -I 



V 2' and 2 r 



vii = m2 = mi = nia = ^ " 



where 12 = + OT2<*^2) when n is large. These approximations are 

 shown to involve logarithms of wi, wg, and (wi + 0)2) ; and the relative 

 distribution of the points in the Argand diagram representing these 

 quantities causes the introduction of two numbers on and m' of 

 fundamental importance in the theory. The double gamma function 

 r2(,2;/wi, 0)2) given by 



is shown to satisfy the two difference relations 



r,-i (^+(02) ri (^/(oi) 2».'^.-s/(.M) 



The functions 721(01, 0)2) and 722(^1, (02) are called the first and 

 second double gamma modular forms respectively. 



The double gamma function can be expressed in two ways as an 

 infinite product of simple gamma functions ; it can be connected with 

 an unsymmetrical function G (z/r) first considered by Alexeiewsky ; 

 and in terms of it, Weierstrass' elliptic functions can be expressed. 

 By means of the values of the numbers m and m' the well-known 

 relation 



171(02 — 172(01 = ± ^Tri 



esLii be obtained, as well as the fundamental formulae of the a- function. 



Fundamental in the theory of double gamma functions is the double 

 Eiemann ^ function, ^2(s, ft/wi, ^2), which is considered in Part III, and 

 is the simplest solution of the difference equation 



f(a + + (02) -f{a + (oi) -f{a + (o.) +f{a) = ^ , 



where s, a, to, (oi, and (oo have any complex values such that (02/(01 is not 

 real and negative, and cr^ has its principal value with respect to the 

 axis of - ((oi + (02). This function is expressible as a contour-integral 

 by means of the relation 



