DOUBLE riAMMA FUNCTTON. 
85;:^ 
We have then, in the hniit wlieii n is iiitinite, 
r = 0 s = 0 \ 
ill ii + //I — I "I 
• 1 - 
n.n'o 
*- W2 
m 
— — nl^ + 1 
X Exp. 
/ , \ "l + "3 /v ’W' 1 \ . 1) (2™ — 1) / O , ox 
- m {m - 1) — t - - y,,j + - - -(ojp + oj,-') 
\ 0 
m.2 
(m - 1)2 
+ - ^ W]^Ct>o 
-I 1 
■V V' -- 4_ 
li/.-y 
Utilise now the extension of Stirling’s theorem and the limit formulas for 
S and S S' Tw. We find that 
0 O 0 0“ 
, ’W ‘ TH -1 / ''"i + '5"3\ 0 0 o o o , T 1 
loo’ n n r., M — - -= {— m'ii~ — 2«G?i — m- + Ij log ni 
r=0 s=0 ' V 
//I 
(. I "’'ft 
fi- (mn fi- m — 1)^ log (mn + m — 1) — f (mn + m — 1)' 
-f (mii. fi- m — 1.)^ Eo [.oS/®^ (co) oj log w] + 2 (nm + w — 1) log (mn fi- m — 1) 
— 2 (mn + m — 1) fi- (nm + m — 1) Eo [38/*^ (cu) co log w] 
+ [1 — oS]' (o) ] log (mn + m — 1) + (1 — nr) log , wo) + Eo [oS/ (co) log co] 
— rn~n~F., [oS/®^ (oj) co log co] — 2 m‘^ \ii log n — ii\ 
— ni'ii Ee [oS/®^ (co) CO log co] — m~ [l — .,8/ (0) ] log n — m-Y., [oS/ (co) log co] 
CO I + COo 
— m~ 
o 1 'hd' 
?r loo' n — 
o *} 
— m (m — 1) ‘ “ I oSj® (0) 2 (rn + ni) ttl fi- iiFo [oS/®’ (co) co log co] — 38 /^' ( 0 ) log 7 i 
+ ^3 [ 38 ,^^ (oj) log co] 1 + 
2iv'~‘ — A 1 / 0 1 o\ I — 2'Oc A 1 
(cop + cop) + - C 0 jC 03 
12 
X - 2 (m + m')7rc . 8 /®' ( 0 ) + - log n - [ 38 ^® (co) log 
CO 
CO^CO.T 
where the logarithms have their principal values with respect to the axis of 
— (cOj + COo). 
The labour of reducing such an expression is evidently very great. It is diminished 
by observing that the result must be independent of n, so that we may neglect all 
terms which involve tills letter ; but even then it is only after several steps that we 
prove that the right-hand side is ecpial to 
[1 - 38 / ( 0 ) ] log m + (1 - ad) logp 3 (co^, co^) 
— 2 (rn rn) ttl 
, 0 + “d" 2 od — 3ai + 1 cod + co.r ad — 2oc + 1 
- {nr — m) \ - ~ + 
12 
COiCOo 
4a) j^a)o 
= [1 — 38 / ( 0 ) ] log w -h (1 — 'in?') log p 3 (coj, COo) + (n? — 1) 2 (m -h m') 77108 ^ (0) 
VOL. CXCVI.—A. 2 Z 
