314 
ME. E. ME BARNES ON THE THEORY OF THE 
For all other values of oj^ and u., the function To ^ | cuj, w.,) exists. But the 
product 
Ur^(21 db i 
and consequently cr ( 2 ), will not exist when either Wo/oj^ or — is real and 
negative ; that is, when r is real. The criterion for the existence of multiple gamma 
functions (/^-ple where n is greater than 2) is more intricate, and, as we know, 
7i-ply periodic functions [n > 2 ) do ]iot exist. 
Part III. 
Contour Tntofirah connected with the Doidjle Gamma Function. The Double 
Riemann Zeta Function. 
§ 37. In the theory of the simple gamma function it was shown that the interven¬ 
tion of a definite integral, coupled with tlie theory of asymptotic approximations, it 
was possible to ol)tain contour and line integrals to express Euler’s constant y, and 
the logarithm of the simple gamma function and its derivatives. We now proceed 
to show that it is possible to extend the method thus previously employed so as to 
obtain expressions as contour and line integrals for the gamma modular constants 
y^j and yoo, and the logarithm of the double function and its deriv^atives. It will be 
noticed that when the real parts of and Wo are positive, the numbers m and m' 
which intervened in Part II. vanish, and there is consequently a noteworthy simplifi¬ 
cation of the formulae olRained. This simplification extends also to the definite 
integral expressions, and consequently we shall first investigate the theory in this 
simple case, proceeding subsecjuently to contour integrals of greater complexity. 
Finally we make use of an extension of Merlin’s method of defining the simple 
^ function by a series Instead of a contour Integral, and we show that there is 
com})lete agreement between the formulae obtained in the difterent ways. 
§ 38. When the real parts of and Wo are positive, and when in addition the real 
part of a is positive, we define the double Riemann ^ function 
i, (.s, a I Wo) 
for all values, real or complex of .s', by the integral 
HRl — s) r z)^-'^dz) 
‘Itt J(1— (1 _ ’ 
wherein r) ''-'log (— 2 ) being real when s is negative and being 
rendered uniform by a cut along tlie positive direction of the real axis. The 
integral is to he taken along a contour enclosing the origin (hut no other pole of the 
* The existence-ci'iteriii for functions which are snljstantialh'^ nniltiple gamma functions have been 
discussed l>y CiiANi, ‘ Batt. Gior.,’ vol. 29. 
