174 MAJOE P. A. MACMAHON: MEMOIE ON THE 



and that C s involves the factors 



( ]( _ t] a^a 



a 



a, 



. 



a t t+n a t + stl 



vt f v ) in ^^t I " ** - 1 - 



Hence a t C t + a,C, involves a factor which by elimination of a t by means of the 

 relation 



1 ^ = 



a t +s t 



may be written 



( 1) ((_! a s + n t-+ 1) (a,_i a s + 1) (o s s + ti) (! , + } (a, a s+1 ) (a, a (+1 s + <) 





and this is at once seen to be zero. Hence 



contains 



1 2t_ (s>t) 



a t +st 



as a factor. It therefore contains C as a factor. 



It also contains another factor which is linear, and considerations of symmetry 

 show it to be 



Hence the expression assumed for 



satisfies the difference equation, and is evidently the expression corresponding to the 

 problem in hand. 



25. We now observe that a redundant generating function, viz., an expression for 

 the sum 



^? / /-j /v // ft \ /y ^i /y ^"2 ^y* ^3 o "* 



^t^]j ^2* ^3) n )/ *^1 *^2 "^"3 * *^n j 



is 



n" ( n"Yi--^ 



=t+l t=l \ X 



l (x 1 + x 2 +x a +...+x n ) 



The question in probabilities, here, I believe, solved for the first time, may be 

 stated as follows : 



