CONNECTED WITH GAMES OF CHANCE. 157 



sequence becomes 



/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.-^'}; 



f{a,b) must therefore equal unity, and/, (a, 6) must vanish 

 when a is an even number. This gives 



u 



{2 + 2^ 1=:^* +2^ 1^1^*/ K*)}. 



This expression is reduced to 2u \i b h an even number, 

 and to 



4m + 4m/ {a,b) 



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

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



even, equal zero ; ~g~ " is such a function, and we then 



have for the third stake 



» { 2 + 2 1=^' + 2> i=^M=^- } . 



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, 



We 



