326 Lord Rayleigh : Problem of Random Vibrations, 



probabilities of the plus and minus alternatives are equal. On 

 account of: the symmetry, no odd powers of x can occur. 

 I have calculated the resulting expression with retention of 

 the terms which are of the order Ijn 2 in comparison with 

 the principal term. The resultant x itself may be considered 

 to be of order not higher than s/n. 

 By Stirling's theorem 



nl=\f(2Tr)n n + h e- n C n , .... (11) 

 where (^1 + ^+-^+..., . . . (12) 



the moment we omit the correcting factors C. Thus 



For the logarithm of the product of the two last factors,, 

 we have 



n + l j x~ a, A .?; G \ _ if_ _ a* _^ 



2 \n 2 + 2n 4 " f 3w e+ * , *J n '6n* On* "" 

 _aP x- tf* /l 1\ ff 6 /l 1\ 

 2n + 2u" 4n*\3 n) 6n 5 \5 n/ "•' 



and for the product itself 



- x >2nj r , l/* 2 **\ , 1 At 4 3./- 6 x s \\ , 10 J 



The principal term in (10) is 



— ¥ '^) e =vwr • 



There are still the factors to be considered. We have 



(Ji( n _. r) V^i 



]i (n _,)Ci( K+I ) l 12?2 28bVJ 



r l l I' 1 U 1 l )-i 



{ + 6( n -«) + 72(?i-«r) 2 j ( + 6(n + tf) + 72(w + a?)*j 



= l l+ is + «jl 1 ~3n" f ^ 2 (rwj 



= i -^ + ^?( i - 3 £ ! ) ( 14 > 



