CONNECTED WITH GAMES OF CHANCE. Ill 



the nth part of the sum of these three profits added to u, and 

 multiplied by ( — l) d , gives for the profit on the fourth event 



u (_ i)- + I ((-1)« (_ 1? + (- 1)* (- 1)' + (- l) c (-1)*) 

 -f> ((-1)" (_!)* (-l) rf +(-l) a (-l) c (-l) d +(-^) 6 (-l) c (-l>) 



+ i (-i) a (-ly (-i) c (-i) rf - 



From these expressions, continued a few steps further if ne- 

 cessary, it appears that the sum of all the profits W is equal 

 to 



u multiplied by the sum of all the quantities 



(— 1)% (-1)*, (-l) c , • • 

 + ~ multiplied by the sum of all the products, two by two 

 of those quantities. 



-f- \ multiplied by the sum of all the products, three by 



lb 



three of the same quantities. 



-f- - 3 multiplied by the sum of all the products, four by four 



of the same quantities. 

 &c. &c. 



Hence - W = the sum of all the co-efficients, except the 



first of the terms of an equation, whose roots are *~ ' - ^~ / , 



r~^ f &c multiplied by u ; but by the conditions of the pro- 



y 2 blem 



