WARING. 449 



(1 + ^)'""^ = (1 + ZY' + ^^ (1 + ^)"^"-'+ i^t^ ^2 (1 + ^)2n-4 



If we exjDand the various 230wers oil+ z and equate the coeffi- 

 cients of z' we shall obtain the arithmetical theorem with t in 

 place of r. 



But it is not obvious how Waring intended to deduce the 

 theorem on the Duration of Play from this arithmetical theorem. If 



we put - for z we obtain 

 ^ a 



{a + hf'^' = a'{a + hf' + ta' {a + by''-' ah + H^A a' (a + 1)'"-* a'l/ 

 + ^ (^ + ^) (^ + ^) a} (a + Z.)^"-^ a^Z.^ + . . . 



and it was perhaps from this result that Waring considered that 

 the theorem on the Duration of Play might be deduced ; but it 

 seems difficult to render the process rigidly strict. 



833. Waring gives another problem on the Duration of Play ; 

 see his page 20. 



If it be required to find the chance of ^'s succeeding n times as 

 oft as ^'s precisely : in ?i + 1 trials it will be found 



in 2n + 2 trials it will be found 



P^n{n + \)^^^^,r.Q; 



in 3/1 + 3 it will be 



^ n {n + 1) (?>n + 1) a'"6' 



V ■• ^ ■/.. . Z,\3n + 3 • 



2 {a + by 



Waring does not give the investigation ; as usual with him 

 until we make the investigation we do not feel quite certain of 

 the meaning of his problem. 



The first of his three examples is obvious. 



29 



