MONTMORT. 123 



posed a game of pure chance, or wliicli is the same thing, the 

 jjlayers are all supposed of equal skill. 



Montmort himself in the case of three players states all the 

 required results, but does not give demonstrations. In the case 

 of four players he states the numerical probability that the money 

 ■will be won in any assigned number of games between 3 and 13 

 inclusive, but he says that the law of the numbers which he 

 assigns is not easy to perceive. He attempted to proceed further 

 with the j^roblem, and to determine the advantage of each player 

 when there are four players, and also to determine the pro- 

 bability of the money being won in an assigned number of games 

 when there are five or six players. He says however, page 320, 

 mais cela m'a paru trop difficile, ou pltitot j'ai manque de courage, 

 car je serois stir d'en venir a bout. 



210. There are references to this problem several times in 

 the correspondence of Montmort and Nicolas Bernoulli ; see Mont- 

 mort's pages 328, 34^5, 350, 3G6, 875, 380, 400. Nicolas Bernoulli 

 succeeded in solving the problem generally for any number of 

 players ; his solution is given in Montmort's pages 381 — 387, and 

 is perhaps the most striking investigation in the work. The 

 following remarks may be of service to a student of this solution. 



(1) On page 386 Nicolas Bernoulli ought to have stated 

 how many terms should be taken of the two series which he gives, 

 namely, a number expressed by the greatest integer contained 



in — . On page 330 where he does advert to this point 



he puts by mistake — instead of . 



(2) The expressions given for a, h, c, ... on j^age 386 are 



2 



correct, excej)t that given for a ; the value of « is ^ , and not 



■^ , as the language of Nicolas Bernoulli seems to imply. 



(3) The chief results obtained by Nicolas Bernoulli are stated 

 at the top of page 329 ; these results agree with tliose afterwards 

 given by Laplace. 



