418 TREMBLEY 



m — 1 games out of m, deduced from the observed fact that he has 

 won n — h games to ^'s n —/games. This chance is, by Art. 760, 



m (n -/+ 1) j^ 



It is needless to go farther, as the principle is clear. The final 

 result is that the fraction of the stake to which B is entitled is 



[ ^-^ ^w+/-l 1.2 n-\-f-ln+f-2 



(/+ 7, - 1) ... (A + 1) {n -/+ 1) {n -/+ 2)...(n-l) 



/-I (^+/_l)(,^+/_2)...(n + l) 



This process is the most interesting in Trembley's memoir. 

 Laplace does not reproduce this problem in the Theorie . . . des 

 Prob. 



772. Trembley gives some remarks to shew the connexion 

 between his own methods and Laplace's. These amount in fact 

 to illustrations of the use of the Integral Calculus in the summa- 

 tion of series. 



For example he gives the result which we may write thus : 



j) + l lp + 2'^ 1.2 p + 3 1.2.3 p + 4<'^"' 



p + q + 1 



==! X^{1- txydx = -^ f'x^ (1 - X^dx. 

 Jo *" J 



773. Trembley remarks that problems in Probability consist 

 of two parts ; first the formulae must be exhibited and then modes 

 of approximate calculation found. He proposes to give one ex- 

 ample from Laplace. 



Observation indicates that the ratio of the number of boys 

 born to the number of girls born is greater at London than at 

 Paris. 



Laplace says : Cette difference semble indiquer a Londres une 

 plus grande facilite pour la naissance des gardens, il s'agit de deter- 

 miner combien cela est probable. See Hist de V Acad. .,, Paris 



