CONNECTED WITH GAMES OF CHANCE. 165 



( — l) a X ( — 1)*, or ( — l) a + b ; observing this, the fac- 

 tor multiplying — | will be expressed thus : 



a c -\- b c 

 ad-\-2bd-\-cd 

 a e -f- 26e + 2c e -J- de 

 af+ 2bf + <2cf + 2df + ef 

 ag + 2bg + 2cg + 2dg + 2eg + fg, 

 &c &c 



If to these were added the two series ab-\- b c-f- c d -f &c. 

 and a(6 + c-f-rf + e+ &c.) the sum would be equal to twice 

 the sum of all the products, taken two by two of the quantities 

 a, b y c, &c. (and since these quantities represent ( — 1 ) a , 

 ( — l) 6 , &c. we have found in the last problem that it is 



equal to / vP~g) ~(P +g) \ . The part depending on v is 

 therefore 



-| { (p-vY - (pH) - (-i)«+*— (-i)»+«_ (-1)'+'- &c. 



-(-!)" ((-l)'+(-l)'+(-l)'+&ft) } 

 The second of these series is evidently equal to 



(-If (p-q- (-1)*) = (-1)° (p-q) -1 ; 

 the other series 



cannot be determined merely by knowing how many of the 

 numbers a, b r c, &c. are odd, and how many are even. 



Still, 





