206 ON THE SOLUTION OF EQUATIONS 



We may deduce from this formula, by indirect considera- 

 tions, one or two analytical theorems. For it is obviously im- 

 possible that the party should terminate in M 's favour in less 

 than x games, as x is the number of counters he must win 

 from N. Consequently 



T TX f T \ Z 



S/ 1 " 1 sin -TT sin TTJCOS -TT) =0 (16). 



a a \ a J 



for all integer values of z less than x 1. 



Again, M may win the party at the 03 th game, if he win x 

 games in succession, the probability of which is p x . Hence, 

 putting z = x 1, we have 



p*= - UpqY f^V 2"" 1 sin - TT sin ir( cos - wY ' 

 a v \27 a \ a j 



or Sj ' 1 sin - TT sin TT^COS - TTJ =^ (17). 



These formulae may undoubtedly be established by other methods, 

 but I have thought it worth while to point out this way of 

 deducing them, from the analogy it bears to that in which many 

 remarkable theorems are obtained by Poisson, in his Theorie de 

 la Chaleur, namely by considering the nature of the quantities 

 which his formulas represent. This mode of establishing analy- 

 tical theorems by considerations founded on the interpretation of 

 our results, is one of the most curious features of the more recent 

 methods of treating physical questions. 



To (16) and (17) another theorem may be added, by the 

 following consideration. Jf, if he win, must win the party 

 either in x games or in x + and even number of games. For 

 if he lose k games he must win back Jc games and x more or 

 there must have been x + 2k games in the party. Hence his 

 chance is zero whenever z + l=x + 2k + l, and therefore 



s, 



a-1 



r rx f r \ x + 2k 



sin - TT sin TH cos - TT) =0 (18), 



a a \ a J 



k being any positive integer whatever. 



When a is infinite, the sums contained in the last three 

 equations become definite integrals. Let 



r j x-u *r * *. z a ~ 1 



- TT = 9, then - = a<f>, and - - TT = TT. 



a a a 



