10 ON A DOUBLE INFINITE SERIES. [102 
which proves the identity of G m with the function T (to). {Substituting in two of the 
preceding equations, we have 
TnlV... A r (n + n'...) = ^ C w “ *] r l n ' ~ ^ > • • ■ ( 8 ) 
(where,as before, p denotes the number of terms in the series n, n',... and r+r'+...=k+l—p), 
the first side of which equation is, it is well known, reducible to a multiple definite 
integral by means of a theorem of M. Dirichlet’s. And 
F O’ n ) = |- TO + 1 jt cpn+n-i-k Sr [m - l] r [n - 1]^-% (9) 
where r extends from — x to + x, and k is arbitrary. By giving large negative 
values to this quantity, very convergent series may be obtained for the calculation of 
F (to, ??)}.