o 
• ) 
54 
ME. E. W. BAENER ON THE THEOEY OF THE 
We thus see that 
Hi 1 ?/l I / \ 
n n' r .( V / ' ) = [p. (wi, wo ).'m =^oo)-2(«i+ 
,■ = 08 = 0' 
III' 
§ G4. In tlie case when m = 2 the preceding result has an especial importance. 
It is convenient to write 
and in accordance witli this notation we put 
These functions evidently correspond to the functions 
cr{z), 0-1(2), 0-0(2), 0-3(2), 
in Weierstrass’ theory of elliptic functions. 
Emitting the zero argument, we take 
I Wi, ojo) = yi (<^1, Wo) and two similar et[nations, 
so til at 
" ' “ 1 /!’&) + So)., 
n n' r, ~ - 
}■ = 0 s = 0 ~ \ 4 
= 7172 73 ^ 
the parameters wi and Wo being omitted. 
And now, from the result of the preceding paragraph, 
71 72 73 = e-3. 2 ...NO) 
so that p. (wi, 0 ).^ = ^(yiyoyg) 2» 
•>5 [1 - .Si' ( 0 )] .. ’S,' ( 0 )^ 
We thus express the double Stirling function of Wi and wo in terms of the product 
of double gamma functions whose argument is a half quasi-period. 
We have previously seen in the theory of the simple gamma function that 
pA<->)= 2 ’ri 
and the formula just obtained is the natural extension of this result. 
§ Go. From the results of G2 and (EG we see that we may express the multiplica¬ 
tion formula for lh( 2 ) in the form 
