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 =i 



n ' 



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

 ment of (x + D' (x — i)^ = 0, 



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

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



iw = (i + i)'(.-D'»-». 



or W = n('-±iyC^yu-«u; (6) 



if pzz q this becomes 



W=^^^^u-nu (7) 



if M rr 1, whatever be the numbers p and q, the loss will be 

 equal to u (unless at the same time ^ — 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 wth part of all his former loss, which since 

 n rz 1 is — u, his next stake must be m — u, 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 u, and if he 



draw 



