CONNECTED WITH GAMES OP 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' 

 into ( — 1)", 6 into ( — 1)*, c into ( — \)\ &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 u, is 

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

 roots are ( — 1)% ( — 1)\ (— 1)% &c. or of the equation 



but we know that p of the quantities a, 6, c, &c. are even 

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

 f equal roots of the form -|- 1, and q equal ones of the form 

 — 1, or the equation is 



o^{x-VY (^+ir; 



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

 of the third term of this expression, which is 



jP-P — 1 Z 9 _L g-9— 1 _ (P— g)' — (p + g) . 



1.2 I'lf 1.2 — 2 



so that the profit of the gamester is 



-^ -{p—q)u — v ^P-iy-^P + i\ (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 (—])", (—1)', [~l)\ &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 Jq 



