PASCAL AXD FERMAT. 19 



Let M denote the number of cases which are favourable to A , 

 and N the number of cases which are favourable to B, Let 

 r = 2/1 - 2. 



In the second example we have 



M — N.= . -^ = X say. 



\n— 1 I ;2 — 1 "^ 



Then if 2 aS' denote the whole sum at stake, A is entitled to 

 -^ . — ^— , that is to — (2*' +X)\ so that he may be considered 

 to have recovered his own stake and to have won the fraction 

 ^7 of his adversary's stake. 



In the third example we have 

 il/ + lY = T-\ 



2 r - 1 2 (?2 - 1) 1 r - 1 1\{n-\\ 



n — \ ?i— 2 \n — 1 In — 1 



Thus we shall find that A may be considered to have recovered 

 his own stake, and to have won the fraction ■— j of his adversary's 



stake. 



Hence, comparing the second and third examples, we see that if 

 the player who wins the first point also wins the second point, 

 his advantage when he has gained the second point is double what 

 it was when he had gained the first point, whatever may be the 

 number of points in the game, 



25. We have now analysed all that has been preser\'ed of 

 Pascal's researches on our subject. It seems however that he had 

 intended to collect these researches into a complete treatise. A 

 letter is extant addressed by him Celeberrimce Matheseos Academice 

 Parisiensi ; this Academy was one of those voluntary associations 

 which preceded the formation of formal scientific societies : see 

 Pascal's Works, Vol. iv. p. 356. In the letter Pascal enumerates 

 various treatises which he had prepared and which he hoped to 



