DOUBLE GAI\IMA FUNCTION. 
307 
CO, 
P — ( \ T ^’i 
a — — 7.33 (wx, W3) + - — 
we find on making 2 = 0, 
— 723 ("n "2) = + U "i ~ 7)’ 
and therefore 
Differentiatino’ ao-ain, we have 
I OJj, W,) = 2A + 2 + ^^20^2 I "1) “ '^/'^('''U^2 1 ^1)] 
Again, making t = 0, we find 
ft), 
ft), 
Wo - 1 
or 
— 7.31 (ftjj, Wo) = 2 A, 
A = — 1-7.03 (w,, Wo) . 
Finally, then, we have 
ro(2 1 wj. Wo) = e - ' U-- u,,' ) X r(':;lw3) 
X n 
i/?.> = 1 
li(.i -f* ^/?.oft)., I ft),) — r,/,, (') I wj) — " ,/,i(2)(r,,ou.i I w,) 
-!- (, ■> 
Ih (//(oft)o I ft)i) 
Thi.s formida is equivalent to the one obtained in the “ Genesis of the Double 
Gamma Functions,” § 2. 
It is an interesting verification to actually transform the one formula into the 
other, making use of the relations e.stabllshed between 7oi(w3, w.,), 7.o.o(w3, Wo), G(t) 
and D(t). 
On account of the symmetry of the ]:)resent functions, the formula corresponding 
to that given in § 8 of the “Genesis of the Double Gamma Functions” may he 
written 
r.T {z ] Wj, Wo) — € 
- V-.1 ~ I Y22 + T log t 
- n ti ' w.> 
V 
• Fi (2 i Wo) 
X II 
m, = 
GG + 7/qft)j I ft)o) 
G (nqft)^ I ft)o) 
— -'I<[(’l0”i“i I “ 2 I 
c 
V I Wa) 
The product formulae just obtained correspond to the expression of the o- function 
as an infinite product of circular functions. 
Such a circumstance, of course, at once prompts us to try and find a fornnda 
corresponding to the expression of the cr function as a mim of exponential functions. 
But it Is readily seen that such is an impossibility. We cannot express the double 
gamma function as a sum of an Infinite series of simple gamma functions of varying 
arp’iiments. 
o 
It Is this fact, combined with the absence 01 any quasi-addition theorem for the 
double gamma functions, which precludes the possibility of any collection of formulae 
rivalling in number and elegance those of the doubly periodic functions. 
2 R 2 
