CONNECTED WITH GAMES OF CHANCE. 173 



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

 ball be marked two, he receives n 2 times his stake, and so on. 

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

 to have been marked i t he adds to the last stake the sum vni : 

 any of the numbers n 1 n 2 , . . may be negative, if he has drawn 

 p balls marked w,, q marked w 2 , r marked w 3 , and so on, 

 what is the amount of his winning ? 



Let a, /3, y, . . . be the kth roots of unity, the expression 



k 



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

 let 



P a — n l S« + n 2 S a +i -f n 5 S a+2 + . . + n k S ffl+jt _i ; 

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



»!, »«» »3» * « ' n k' 



With the aid of these considerations, we can express the 

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

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

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

 of this first determination, he adds to u the quantity t;P a , 

 which sum u + v P a forms his second stake ; the number of 

 the second ball determines the amount of his second profit, 

 which may be expressed thus : 



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

 be u + v P a -J- v P* and the profit arising from it is 



that in the fourth event is 



the 



