4G8 LAPLACE. 



approximation arrives at tlie conclusion that this probability does 

 not differ much from unity. 



872. Laplace proceeds to the Problem of Points. He quotes 

 the second formula which we have given in Art. 172 ; he says that 

 it is now demonstrated in several works. He also refers to his 

 own memoir in the volume of the Academy for 1773 ; he adds 

 the followinof statement : 



...on y trouvera pareillement line solution generale du Problcme 

 des partis dans le cas de trois ou d'un plus grand nombre de joueurs, 

 probleme qui n'a eucore ete resolu par personno, que je sache, bien que 

 les Geomutres qui ont travaille sui* ces matieres en aient desire la 

 solution. 



Laplace is wrong in this statement, for De Moivre had solved 

 the problem ; see Art. 582. 



873. Let X denote the skill of the player A, and 1—x the skill 

 of the player B ; suppose that A wants f games in order to win 

 the match, and that B wants h games : then, if they agree to leave 

 off and divide the stakes, the share of B will be a certain quan- 

 tity which we may denote by (j) {x,f, h). Suppose the skill of each 

 2)layer unhioiun; let 7i be the whole number of games which A or 

 B ought to win in order to entitle him to the stake. Then Laplace 

 says that it follows from the general principle which we have given 

 in Art. 869, that the share of B is 



I ic"'-^ (1 - x)""-^ (f) (x, f, h) clx 



J 



1 



x""'^ {y - xY^ dx 



The formula depends on the fact that A must already have 

 won n —f games, and B have won n — li games. See Art. 771. 



874. Laplace now proceeds to the question of the mean to be 

 taken of the results of observations. He introduces the subject 

 thus : 



On peut, au moyen de la Theorie precedente, parvenir a la solution 

 du Problcme qui consiste \ determiner le miUeu que Ton doit prendre 



