448 WARING. 



a"- f ah r (r + 3) a%' r (r + 4) (r + 5) a'6' 



{a + by i {^tT+Vf "^ 2 {a + by ^ "~ [3 (^^« 



+ H — ^ ^^-^ n / TT^ + m infinitum >• 



\l_ (a + by'' j 



This may be deduced from the subsequent arithmetical theorem, viz. 

 2m{2m-l) (2m -2)... (2m -s) (2m- 2)(2m ~ S)...(2m - s - 1) 



— - f. , 



+ 1 [s 



r (r + 3) (2m - 4) (2y?i - 5) . . . (27;t - g - 2) 



1 



^(^+4)(r + 5) (2m-6)...(2m-s-3) 



+ 's-2 



+ ... 



+ 



T (r + s + 2) (r + s + 3) . . . (r + 25 + 1) 



+ 1 



(r + 27??.) (r + 2y?z, - 1) ... (r + 2y?i - s) 



s+1 



Waring's words, "^ happening r times more than B" are 

 scarcely adequate to convey his meaning. We see from the for- 

 mula he gives that he really means to take the problem of the 

 Duration of Play in the case where B has a capital r and A has un- 

 limited capital. See Art. 309. 



Waring gives no hint as to the demonstration of his arith- 

 metical theorem. We may demonstrate it thus : take the formula 

 in Art. 584, suppose a = l+^, ^ = 1, ^ = ^; we shall find that 



Thus we get 



^- (i+zy'^' {1 + zy'''^ 2 (i+^r 



^ (^ + 4) (^ + 5) z 



U3 (1 + z) 



t+G 



+ [4 {1 + zy^''^"" 



Multiply both sides by (1 + z^"^' : thus 



