CONNECTED WITH GAMES OF CHANCE. 161 



the roots of an equation, then the expression just written is 

 equal to the co-efficient of its third term. If we change a' 

 mt0 (—1)", I) into (— l) 5 , c into ( — l) c , &c it becomes the 

 same as the one whose value we are seeking. Hence then it 

 appears, that the sum of all the quantities, multiplying v 9 is 

 equal to the co-efficient of the third term of an equation whose 

 roots are (— l) a , (— l) 6 , (— l) c , &c. or of the equation 



o = x — (— 1)-.*— (— l) b .x— (— l) c 



but we know that p of the quantities a, b, c, &c. are even 

 numbers, and q of them odd ones ; therefore this equation has 

 p equal roots of the form -j- 1, and q equal ones of the form 

 — 1, or the equation is 



o= [x—l) p (*+l) ? ; 



and the quantity which multiplies v, is equal to the co-efficient 

 of the third term of this expression, which is 



p-p— i p 9 _l 9-g — i _ (p — g) 2 — (p + q) . 



1. a 1 • 1 "r" 1.2 — 2 



so that the profit of the gamester is 



W=(p-q)u-v ( P-Qr-<P + <1) . (1) 



This result is entirely independent of the order in which the 

 events occurred ; and we may learn from the method that has 

 been employed for its solution, that whenever the sum of all 

 the winnings or losings is a symmetrical function of the quan- 

 tities ( — l) a , ( — l) 6 , (— l) c , &c. the final conclusion will 

 not depend on the order in which the events succeed each 

 other, but on the actual number of favourable and unfavour- 

 able events. 



vol. ix. p. i. x In 



