DOUBLE GAMMA FUNCTION. 
347 
+ nSj [« + {im + l)w_^ + {qn + log' ['"<■ + 1-)"J 
~ "I" (P^^ “1“ l)"i] H~ “1“ l )*^-^]] 
“ “1“ H“ l-)"J10o H“ “t“ 1)^:.’] 
+ 
4cdi&)o 
a 
O 
W, + Wo 
WlWo 
' -T,{pn + l)0pi+ 1) 
ill the limit when n is made iiihnite. 
Substituting the value of oS/(a) in the various terms, expanding tlie logarithms, and 
re-arranging the result in powers of n, we find, with the symbolic notation of ^ 49, 
* = 0 
pn qn 
= — log n n (a + n) + pq\ji~ log n — n-({ + 1)] + {p -h q) [/; log n — 
0 0 
mi 711.2 
+ ^Fo[2S/®>((y) (2:)w)~log^9w] + 5iFo[2Sf~^(a + (o)pxxtlogpoP\ 
-|- F2[oS^'(rt + &))log23w] + [1 — (''0] l^g 
in the limit wiien n is made infinite. 
Since the left-hand side of this ecpiality is l)y the definition of «|wj, w.,) finite 
unless a is at one of the points 
/??! = 0, 1, 2, . . CO I 
m.^ = 0, 1, 2, . . ., GO J ’ 
we see that we have thus been led in a purely algebraical manner to the determina¬ 
tion of the dominant terms of the fundamental exjoansion of ^ 49. 
If we make a = 0, and rememlier the definition of § 50, viz. :— 
^ i ^2-^ 2®1 
S 
— log p.i (wj^, wj) = 
.5=0 
a = 0 
(S, Cl- I Wp W,) nS]^ (c) . 
--— + log a 
we arrive at the dominant terms of the extension of Stirling’s theorem to two 
parameters. 
If we utilise this result in conjunction vdtli the one just obtained, we find 
U 
s = 0 
■?2(S/0 - nS/fft)' 
~ log 
'III i'll’ 'll’- na 
= — log a -f- log 11II' n — log IT n' («-}- n) fi- — F^ [^pw logpt^] 
^00 ^00 ^1^3 
"1“ ^*"3 [{3^0 + w) “ 3^0 (^^)] logp^] — 2^0 (^0 log 
and by the definition of the double gamma function the expression last written 
teduces to 
2 Y 2 
