172 AN EXAMINATION OF SOME QUESTIONS 



blem,^ of the quantities a, b, c, are even, and q of them odd 

 numbers, or p of the roots of the equation are of the form — 



n ' 



and q of the form — , and the equation itself is the develope- 

 ment of (x + ^f (x — }) q - 0, 



the sum of all its co-efficients except the first may be found 

 by making x~ 1, and subtracting unity, this gives 



or W = n(«-±iy(^l)'u-nu; (6) 



if p zr q this becomes 



(n* _ l)p 



if n — 1, whatever be the numbers p and 9, the loss will be 

 equal to u (unless at the same time q =z 0), for in that case as 

 soon as the first unfavourable event happens, the player loses 

 not only all he had previously won, but also the sum u be- 

 sides ; and since, by the conditions of the play, he must sub- 

 tract from u the rath part of all his former loss, which since 

 n == 1 is — w, his next stake must be u — u y or zero; so that in 

 fact if n = 1, the first unfavourable decision terminates the 

 game. 



I shall now proceed to show how a similar mode of reason- 

 ing may be applied to cases where the number of events which 

 happen at each step are more than two. 



Suppose a person draw a number of balls in succession from 

 an urn, containing balls numbered 1, 2, 3, 4, . . k, there being 

 many of each kind : he begins by staking the sum w, and if he 



draw 



