LAPLACE. 531 



Suppose that cj) {x^, x^, x^, ...) is the coefficient of ?f/i t^'^^ts^'s ... 

 in the development of a function u of these variables. Laplace 

 then proceeds thus. From (1) he passes to 



?^ = w(M+M + M+---) (2)' 



and then he deduces 



i=M+M+M + O^)- 



Hence 



therefore 





, ^1 (^1 + 1) 



1.2 



(M + M+--0' 



cc, (a?, + 1) (x, + 2) , , . . 



+ 



Now the coefficient of t^t^^^t^^... in — ^ is ^ (a^^, ajg, cCg, ...)• 



Let 7i:w;:)^*i i^"* t^^ . . . denote any term of the right-hand member 

 of the last equation. Then the coefficient of t^ t,^2 t^a ... in this 

 term will be hp^i <f> (0, x^—m, x^—7i,...). But cj) (0, x^— m, x^—n,...) 

 is equal to unity, for if the first player wants no points he is en- 

 titled to the stake. Moreover we must reject all the values of 

 <f> if), x^ — m, x^ — n, ...) in which m is equal to or greater than x^^ 

 in which n is equal to or greater than x^, and so on; for these 

 terms in fact do not exist, that is must be considered to be zero. 

 Hence finally 



^ (a?j, OJg, a-g, . . .) =2\^' |l + a?, {p^ +793 + . . .) 



* !■ 



34^—2 



