LAGRANGE. 31 



Ht 



without demonstration in De Moivres second solution ; see 

 Art. 309. We will give Lagrange's solution as a specimen of his 

 methods. We may remark that Laplace had preceded Lagrange 

 in the discussion of the problem of the Duration of Play. La- 

 place's investigations had been published in the Memoires . . . par 

 Divers Savans, Vols. vi. and Yii. 



Laplace did not formally make the supposition that one player 

 had unlimited capital, but we arrive at this case by supposing 

 that his symbol i denotes an infinite number ; and we shall thus 

 find that on page 158 of Laplace's memoir in Vol. vii. of the 

 Memoires... par Divers Savans, we have in effect a demonstration 

 of De Moivre's result. 



We proceed to Lagrange's demonstration. 



584. The probability of a certain event in a single trial is^ ; 

 a player bets that in a trials this event will happen at least 

 h times oftener than it fails : determine the player's chance. 



Let y^i represent his chance when he has x more trials to 

 make, and when to ensure his success the event must happen at 

 least t times oftener than it fails. Then it is obvious that we re- 

 quire the value of 7/„^j^. 



Suppose one more trial made ; it is easy to obtain the follow- 

 ing equation 



The player gains when ^ = and x has any value, and he loses 

 when x=0 and t has any value greater than zero ; so that y^^^=\ 

 for any value of x, and y^^t^ for any value of t gTeater 

 than 0. 



Put ^ for 1 —p, then the equation becomes 



To integrate this assume y = ^a'^/3* ; we thus obtain 



p - a/B + q/3' = 0. 



From this we may by Lagrange's Theorem expand /S' in powers 

 of a ; there will be two series because the quadratic equation 

 gives two values of /S for an assigned value of a. These two 

 series are 



