334 
MR. E. BARNES ON THE THEORY OF THE 
For putting = 0 in the asymptotic expansion of 
pn qa 
log II II (o + .a) 
Ml = 0 ni., = 0 
we find 
pn qn 
log H II fi- WoWo) 
m, =0 m, = 0 
= - 
a = 0 L' 
= 0 
ds 
(5, Cl I coy, Wo) “h log a 
+ 2^q [n" log n - + i) 
-f (p + q)\2n\ogn — n\ + ?t-Fo [oS/'^^ (oj) pw logpw] + nFo (w)pw logpw] 
+ Fo [oS/ (w) logpw] fi- log n [1 — oS/ ( 0 ) ] 
+ 2 
m = 1 mn 
(—)“ ^ jpi + 2^m + l 
(yw)" 
(1). 
This is the extension of Stirling’s Theorem to two parameters. If for all values of 
and Wo, we put 
~ 0 
I (s, a \ wj, Wo) + log a 
log po (Wj^, Wo) = - lut 
a = 0 
s = 0 
we may call po (w,, Wo) the double Stirling function of the parameters w^ and wo. 
It is the same as the third double gamma modular form |)reviously defined. For we 
have by § 43, 
log po (Wj, Wo) = ht 
a = 0 
il 
= u 
a = 0 
5 = 0 
e ‘^’-(-z) i{log( —s) + y]ch 
(1 — (1 — 
0 
— { (s, Cl I W|, Wo) — log Cl 
log a 
2M7rtoS/ (0 I w^, Wo). 
by § 42 
We now see the exact analogy between the function po (w|, Wo) and the simple 
Stirling function p^ (w) = .^(27r/w) as defined in § 31 of the “Theory of the Gamma 
Function.” 
For a brief inspection shows us that the result of § 30 of that theory may be 
written 
log II w) = j) [ii log 11 — n) fi- r[Si® (w) j^iwlogp)w] 
•/Uj = 1 
+ [1 4- S'l ( 0 )] log n + log p^ (w) + S/ (w) log pco 
I y ( — Sm(w) + iBffi+i 
m= 1 V/i7l"‘ (7^0))“ 
which is the complete form of Stirling’s Theoi'em for a single parameter. 
The analogy between this asym])totic ex})ansion and 
