CONNECTED WITH GAMES OF CHANCE. 157 



sequence becomes 



. { 2 + 2 " i=Sj=a 7k b)+ 2 s ^jpa-7. &)f-> 



/and /^ not being the same functions as before. If a had been 

 an even number, we may consider the first bet as not having 

 been made, since it has no influence on the succeeding ones ; 

 and in this case the expression ought to reduce itself to 



u 



{2+2 .i^p!)> }i 



f{a,b) must therefore equal unity, and f t (a, b) must vanish 

 when a is an even number. This gives 



u 



{ 2 + 2 ' htkrH • + jf i=£=a V, fc *) } • 



This expression is reduced to 2 m if 6 is an even number, 

 and to 



4« + 4«/ 1 (a t b) 



when b is an odd one ; f x (a, 6) must therefore be such a func- 

 tion of a, that when a is odd, it shall become unity, and when 



even, equal zero j a """ " is such a function, and we then 



have for the third stake 



u 



{* + ^s=i=^+^-=ip±\ i=fc=n-}. 



The law by which we may represent the stake to be ventured, 

 after the determination of any number of events, is now 

 apparent ; had it not been sufficiently so, the same reasoning 

 which has been already explained at some length would have 

 assigned it for the fourth stake, 



„{ 2+2 l=jf=a • +2 'l=fc!»'. l=t=li°+2 1=6=1^1=0=11'. 1=6=12-} . 



We 



