CONNECTED WITH GAMES OF CHANCE. 165 



( — 1)" X ( — 1)*, or ( — 1)''+* ; observing this, the fac- 

 tor multiplying — | will be expressed thus : 



a c -\- b c 

 ad + 2bd + cd 

 ae-{-2be-\-2ce-\-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-j- c d -\- &c. 

 and a(i + c+^ + e + &c.) the sum would be equal to twice 

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

 a, b, c, Sec. (and since these quantities represent ( — 1)% 

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



equal to ^(/P — g)' — (p +9) y The part depending on v is 

 therefore 



-I { ip-qf - ip+q) - (-!)«+'- (-i)H<=_ (_i)c+<^_ &c. 

 -(-i)''((-ir + (-i)°+(-ir+&c.) } 



The second of these series is evidently equal to 



(-1)" (p-q- {-!)") = (-1)" ip-q) -1 ; 

 the other series 



(_l)«+i + (_l)Hc_|. (_l)e+d^ (_l)rf+e_|_ 5j^^ 



cannot be determined merely by knowing how many of the 

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



Still, 



^ 



