in the Theory of Probabilities. 233 



(correct to order l/n 3 ) t 



log{ftn-i*)!}+log{(i/i + i«)!} 



= (n + 1 ) log (in) - n + log (2tt) + ^ 2 /n 



i .?' 4 i ,i- 2 i r 



( l il — l — - 1 L \ 



\12n z %n' + Sn) 



/j 1 ^_l^ 1«*\ 

 + \30 n* 4 rc 4 + 3 /TV 



- \56 y/ 7 6 n 6 ^ 3 n 5 45 n 3 / ' * - ; 



The formula to be evaluated contains also log (n !) which 

 is given at once by writing x=n in (1) 



log(n!) = (n + i)lpgn-n + itog(2 9 r)+ I ^-3g g ^. . "(5) 



If we subtract (4) from (5) and also subtract n log 2, we 

 obtain, on reduction, 



1 / nl ^ 



° g I 2»(in-i«) ! (±n + i*') I J 



1 / 2 \ .r 2 

 = 2 10 «(W"2S 



Vi - — - — X -^ 

 \12 n 3 2 n 2 4 w/ 



/ 1. x l _ 1 ** i t\ 

 \30 n' 4 n* + 3 nV 



\56 n 7 6 n 6 + 3 n 5 ^ 24 n 3 /' * * ' ' 



where the last three lines are respectively of orders l//i, 

 1/n 2 , 1/n 3 . 



This is readily seen to be equivalent to Lord Rayleigh's 

 formula (15) {I c. p. 327). 



In fact call the last three expressions in brackets a, ft, <y 

 respectively ; then, neglecting terms of order 1/n 3 because 7 

 is of order 1/n 3 , 



e~ a ~^ y = l — a + (W-ft). 

 Phil. Mag. S. 6. Vol. 38. No. 224. Aug. 1919. R 



