LAGRANGE. 819 



y... =P' {/(^ -t) + tpqfix - < - 2) + '-^J^^Y/(^ - < - 4) 



the series extends to ^ (a? — ^ + 2) terms, or to ^ (a? - ^ + 1) terms, 



according as ic — ^ is even or odd. 



The other condition is that j/a;,o— ^y ^''^^ ^^7 vakie of x. But if 

 we put ^ = we have yx,Q=f{p^)' Hence f{x) = l for every 

 positive value of x. Thus we obtain 



the series is to extend to 3 (x — t + 2) terms, or to ^ (x — t-\-l) 



terms. This coincides with the result in De Moivre's second form 

 of solution : see Art. 309. 



585. Lagrange gives two other solutions of the problem just 

 considered, one of which presents the result in the same form as 

 De Moivre's first solution. These other two solutions by Lagi^ange 

 differ in the mode of integrating the equation of Finite Differences ; 

 but they need not be further examined. 



586. Lagrange then proceeds to the general problem of the 

 Duration of Play, supposing the players to start with different 

 capitals. He gives two solutions, one similar to that in De 

 Moivre's Problem LXiii, and the other similar to that in De 

 Moivre's Problem Lxviii. The second solution is very remarkable ; 

 it demonstrates the results which De Moivre enunciated without 

 demonstration, and it puts them in a more general form, as De 

 Moivi'e limited himself to the case of equal capitals. 



587. Lagrange's last problem coincides with that given by 

 Daniel Bernoulli which we have noticed in Art. 417. Lagrange 

 supposes that there are n urns ; and in a Corollary he gives some 

 modifications of the problem. 



588. Lagrange's memoir would not now present any novelty 

 to a student, or any advantage to one who is in possession of the 

 method of Generating Functions. But nevertheless it may be read 



