316 LAGl^ANGE. 



This leads to 



yx,t - q |i + f/? + — 2 — P + 273 ' ^ 



+ ... 



(5 + ic- 2 



^ - 1 1 .-r - 1 



This result agrees with the second formula in Art. 172. 



582. The fourth problem is like the third, only three events 

 may now occur of which the probabilities are p, q, r respectively. 

 In a Corollary the method is extended to four events; and in 

 a second Corollary to any number. 



To this problem Lagrange annexes the following remark : 

 Le Probleme dont nous venons de donner une solution tres generale 

 et tres simple renferme d'luie maniere generale celui qu'on nomme com- 

 munement dans I'analyse des liasards le probleme des partis, et qui 

 n'a encore ete resolu complettement que pour le cas de deux joueurs. 



He then refers to Montmort, to De Moivre's second edition, 

 Problem VI, and to the memoir of Laplace. 



It is very curious that Lagrange here refers to De Moivre's 

 second edition, while elsewhere in the memoir he always refers to 

 the third edition ; for at the end of Problem vi. in the third 

 edition De Moivre does give the general rule for any number of 

 players. This he first published in his Miscellanea Analytica, 

 page 210 ; and he reproduced it in his Doctrine of Chances. But 

 in the second edition of the Doctrine of Chances the rule was not 

 given in its natural place as part of Problem vi. but appeared as 

 Problem LXix. 



There is however some difference between the solutions given 

 by De Moivre and by Lagrange ; the difference is the same as 

 that which we have noticed in Art. 175 for the case of two players. 

 De Moivre's solution resembles the first of those which are given 

 in Art. 172, and Lagrange's resembles the second. 



It is stated by Montucla, page 397, that Lagrange intended 

 to translate De Moivre's third edition into French. 



583. Lagrange's fifth problem relates to the Duration of Play, 

 in the case in which one player has unlimited capital ; this is De 

 Moivre's Problem LXV: see Art. 307. Lagrange gives three solu- 

 tions. Lagrange's first solution demonstrates the result given 



