OF FINITE DIFFERENCES. 207 



Consequently (16), (17), (18), become respectively 



I sin</> smx(j> (cos </>)* d<j> = (19), 



J o 



(3 being integral and less than x 1), 



/sin </> sincc< (cos (j>) x ~ l d(j> = ~ (20), 

 j 2 



rsin< sin#</> (cos (j>) x + 2k d<f> = (21). 

 - 



If, instead of seeking the probability that M will win the 

 party at the (z + l) th game, we wished to find that of his 

 winning it after z or more games shall have been played, we 

 should only have to sum (15) for z from z to infinity. Calling 

 this new probability u^ we should thus get 



PO 1 



Wy 



. r . rx 



sin -TT sin TT 



If in (22) we put z equal to zero, we have then the proba- 

 bility of Jf's winning the party at the first, second, &c. 

 games, i. e. of his winning it at all. Writing simply u x for u^ 

 we shall thus get 



. r . nc 



sin - TT sin TT 



Now of this probability we can obtain, as is well known, 

 a much simpler expression. For it is easily seen that we shall 

 have 



u. =pu^ + qu x+l (24) 



for every value of a?, provided that, instead of considering u x as 

 the probability that M will win the party, we make it denote 

 the probability that he either has won or will win it. As it is 

 impossible that he can have already won it while x differs 

 from zero, this alteration does not affect the value represented 

 by u x except for the case of x =0. In this case the value of u x , 

 as expressed by (23), will be zero, as the party is at an end, 

 M having already won it. But according to the proposed 



