CONNECTED WITH GAMES OF CHANCE. 171 



the wth part of the sum of these three profits added to m, and 

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



u (- 1)"+ ;; ((-i)« (- 1)^+ (- 1)* (- 1)'^+ (- ir (-i)') 

 +l{{-\r (-1)' (-i/+(-i)'"(-ir(-ir+(-i)'(-ir(-i)0 

 + I (-1)" (-1)' (-1)^ (-1)^- 



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)*, i-l)\. . 



-f- - multiplied by the sum of all the products, two by two 



of those quantities. 



+ \ multiplied by the sum of all the products, three by 



three of the same quantities. 



+ ^ 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 till" i—^)'' 



^ re ' re ' 



■ ^ > &C. multiplied by u ; but by the conditions of the pro- 



Y 2 blem 



