EULER. 255 



We may write this for shortness thus, 

 {a-+ n A-- + 4;^ A- + " ^"^^;'- '' A-. ...} j^ (0, ,.;}: 



Now put E—1 for A, expand, and reariange in powers of E; 

 we shall thus obtain 



1^" - (/> {n, V + 1) E''-'-' -\-{v + l)^ {n, V + 2) E"-'"' 



- (^^±1)^^ ^ (., . + .3) E-^ + ...} {^ (0, .^ ; 



and this coincides with Euler's result. 



We shall find in fact that when we put E—1 for A, the 

 coefficient of E''~^ is 



(- '^y \J1 U n ^ P^P-^') p{r-V){p-^-.^ I 



\p \n-p \ ^ "^ 1.2 1.2.3 ■^•••j' 



where the series in brackets is continued to z^ + 1 terms, unless 

 p be less than z^ + 1 and then it is continued to jj + 1 terms 

 only. In the former case the sum of the series can be obtained by 

 taking the coefficient of x" in the expansion of (1 — xY (1 - xy\ 

 that is in the expansion of (1 — xY~^. In the latter case the sum 

 would be the coefficient of x^ in the same expansion, and is there- 

 fore zero, except when ^9 is zero and then it is unity. 



460. Since r tickets are drawn each time, the greatest number 

 of tickets which can be drawn in x drawings is xr. Thus, as 

 Euler remarks, the expression 



[<^ («. r)Y - n [4> [n - 1, r)Y + ^^^ {<i> (u - 2. r)]' - ... 



must be zero if n be greater than xr ; for the expression gives the 

 number of ways in which 71 tickets can be drawn in r drawings. 

 Euler also says that the case in which n is equal to xr is re- 

 markable, for then the expression just given can be reduced to 

 a product of factors, namely to 



