THEORY OF PROBABILITIES. 177 



-, 



thenP=- 



r(n) 



Now, by Stirling's theorem, or by that which Binet proposes to 

 substitute for it (vide Journal de I* Ecole Poly technique , xvi. 

 p. 226), we have, when n is very large, 



Consequently 



(w) V[27T (n - 1)} 



being a certain function of 2 and n. Therefore 



Now the coefficient of tf in ft is 



1 



tnat ot t is \ ~ -f" y 1 ^r "ij ^ A > 



\. VI ~" 1 ( 91 ~ 1)J 1 O 



and that of 2* is 



L_l_ _iL_ 6 I J_ 



( w~l (n-l) 2 (w-l) 8 j 1.2.3.4" 

 Hence the coefficient of t* in e~'/ is 



1 



that of ^ is 



111 1 



2(n-l) ' 3(n-l) 3 ^S^-l) 27 



12 



