ON THE SOLUTION OF EQUATIONS, ETC. 203 



conditions than arbitrary functions. There will, generally speak- 

 ing, be an infinite number of arbitrary constants, and it is 

 therefore necessary to treat them in classes. The ingenious 

 synthesis by which this is effected by Fourier, in the different 

 problems discussed by him in the Theorie de la Chaleur, forms 

 one of the most interesting parts of that admirable work. The 

 same kind of reasoning is made use of by Poisson, in his re- 

 searches on similar subjects : and there can be little doubt that 

 the methods of Fourier, developed and extended as they have 

 been by subsequent writers, will long continue to be an essential 

 element in the application of mathematics to physical researches. 

 Similar methods may be made available in the solution of 

 equations in partial finite differences. Such equations do not, 

 it is true, present themselves very often, as the continuity of the 

 causes to which natural phenomena are due, leads rather to 

 differential equations than to those in finite differences. In 

 fact, I am not aware of any subject, except the theory of 

 probabilities, in which we meet with problems whose solution 

 depends on that of an equation in partial finite differences. 



In this theory, however, such problems are not uncommon. 

 One of the most interesting of them, both in its own nature 

 and historically, may serve as an illustration of the application 

 of the methods of Fourier to finite differences. This problem, 

 which has engaged the attention of several writers on the subject 

 of probabilities, and of which a solution was among the earliest 

 efforts of Ampere, is that of the duration of play. Professor 

 De Morgan has spoken of this solution arid of that of Laplace, 

 as being of the highest order of difficulty: that which I am 

 about to enter on has, I think, a decided advantage in this 

 respect. 



The problem itself may be thus stated: Two persons, 

 M and N, have between them a number a of counters : they 

 play at a game at which M' s chance is p, and N'a q. The 

 losing player gives one counter to the other, and they are to 

 play on until one or other have lost all his counters. What 

 is the probability that the party will terminate in M's favour 

 after any assigned number of games, N being supposed to have 

 originally x of the a counters ? 



Let y xz be the probability that M will win the party at the 

 (z + l) th game. If he win the next game (of which the pro- 



