600 LAPLACE. 



Laplace puts for p in succession ^ (1 + a) and ^ (1 — a), and 

 takes half the sum. Thus he obtains for ^'s chance 



1 {(1 + g)"- (1 -^■)"1 [il + oy+{l-of] 



2 (1 + a)"+' - (1 - af^"" 



Laplace says that it is easy to see that, supposing a less than 



h, this expression is always greater than , , which is its 



limit when a = 0. This is the same statement as is made in 

 Art. 891, but the proof will be more easy, because the trans- 

 formation there adopted is not reproduced. 



Put -z. = X, 



1 — a 



and 'ii = . „+6 





We have to shew that u continually increases as x increases 

 from 1 to 00 , supposing that a is less than h. It will be found that 



1 du _ ax" [x^ -l)-ha^ [x"^ - 1) 

 udx" x[x''-l){x'' + l){x''^-l)' 



We shall shew that this expression cannot be negative. 

 We have to shew that 



h a 



cannot be negative. • 



This expression vanishes when a; = 1, and its differential coeffi- 

 cient is {o^~^ — x°-~^) (1 — ic~"~*), which is positive if x lie between 

 1 and oc ; therefore the expression is positive if x lie between 

 1 and 00 . 



Laplace says that if the players agree to double, triple, ... 

 their respective original numbers of counters the advantage of A 

 will continually increase. This may be easily shewn. For change 

 a into lea and h into hh : we have then to shew that 



(a^" - 1) (a^^ + 1) 



