CONNECTED WITH GAMES OF CHANCE. 173 



draw a ball marked one, he receives n^ times his stake, if the 

 ball be marked two, he receives n^ times his stake, and so on. 

 The next stake is thus regulated, supposing the ball last drawn 

 to have been marked i, he adds to the last stake the sum v iii : 

 any of the numbers n, n^, . . may be negative, if he has drawn 

 p balls marked n, , q marked ra^, r marked w,, and so on, 

 what is the amount of his winning ? 



Let «, /3, 7, . . . be the kth. roots of unity, the expression 



S.= «" + ;3^ + y° + .. 

 k 



is always equal to zero, except when a is a multiple oik; also 

 let 



P„ r: Wj Sa + W, Sa+l + W, S„+2 + . . + M/, S„+t_i ; 



then P^ will in every case reduce itself to one of the quantities 

 «., M., n,, . . . n^. 



With the aid of these considerations, we can express the 

 amount of the stake at any particular step ; his first is u, and 

 whatever be the kind of ball drawn, his profit is always ex- 

 pressed by u Pa, according to the form of a. In consequence 

 of this first determination, he adds to u the quantity vVa, 

 which sum u + vVa forms his second stake ; the number of 

 the second ball determines the amount of his second profit, 

 which may be expressed thus : 



{u + v P„) Pi, 



without at all determining the form of a, the third stake will 

 be M + u Pa + f Pj and the profit arising from it is 



that in the fourth event is 



(M + uP.+uPi +t>Pe)Prf. 



the 



