268 ADDITION TO MR GLAISHERS PAPER “ PROOF OF STIRLING’S THEOREM.” [677 
log (l-fd ) 
1 + -J writing e ' \ n !, the whole exponent of 
e is 
We have 
(w+i)log (l + -n-i(a + 6 + ...) 
1 1 _ 1 3 1_ 
n + 3.4 n 2 4.6 n 3 5.8 n* 
— 2 ( a + & + •••)• 
1 . 1 . . 1 1 c 
= const. — —f- terms in — , — , &c. 
4w rr n 3 
l 2+ 3 2+ + (2w+l) 
(the constant is in fact = ¿7r 2 , but the value is not required), hence a = const. — — 
+ terms in —, —, &c.; as regards b, c, &c., there are no terms in -, but we have 
n 2 n 3 W 
b = const. + terms in —„, &c., c = const. + terms in , &c. Hence the whole exponent 
n 2 n d 
of e is 
= — n + C + —b terms in \, &c. 
12?i n 2 
As in Mr Glaisher’s investigation, it is shown that e~ r = V(2tt), and hence neglecting 
the terms in &c., the final result is 
n 2 
_ w -i—L 
n??. = \Z(2Tr)n n+ ±e 12M .