176 AN EXAMINATION OF SOME QUESTIONS 



On compaiing this with the fourth problem which was solved, 

 it appears that 



W —ux the sum of all the co-efficients except the first 

 of the equation (.r + P„) (x + Pj) (•i' + I*c ) • • • = 0, 

 or W=zm(1 + P„) (l + Pj) (1 + Pe)... _M; 

 but J) of the quantities P,,, Pj, Pc, . . . are equal to n,, 9 of 

 them to n^, r of them to n,, . . . This equation, therefore, 

 becomes 



W zzu(l +n,y (1 + nJ' (1 +",)■■. . . — ?< (9) 



if n, = - , n, = — - , n. =0, and u — nu. This coincides 



with (6). 



As an example, suppose an urn filled with balls of three co- 

 lours, white, black, and red, and that the person who draws 

 them out may name any sum he chose prior to each extrac- 

 tion ; if he draw a white ball, the sum he named is paid to 

 him ; if a black, he loses one-half of it ; and if a red one, he 

 loses one-third of that sum. And suppose he regulates the sum 

 named in the following manner, beginning with naming u 

 whenever he has drawn a white ball, he adds the whole of his 

 previous winnings to the sum u ; but if he has drawn a black 

 one, he adds only half his profits to the sum u ; and if the ball 

 last extracted from the urn was red, he adds one-third of all 

 his profits to the same sum u. He has drawn out p white, q 

 black, and r red balls, what is the amount of his profit or 

 loss? 



In 



