176 AN EXAMINATION OF SOME QUESTIONS 



On comparing this with the fourth problem which was solved, 

 it appears that 



W - u X the sum of all the co-efficients except the first 

 of the equation 0+P a ) (a? + P 4 ) + P c ) . . . = 0, 



or W = W (l + P a ) (1 + P 6 )(l + Pc) u; 



but p of the quantities P fl , P 4 , P c , . . . are equal to n„ q of 

 them to w 2 , r of them to n„ . . . This equation, therefore, 

 becomes 



W = m(1+ n t )' (1 + » fl )» (1 + n,)' . . . — u (9) 



l 1 



if n l = - , n x ~ — - , n z = 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 w 

 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 



