OF FINITE DIFFERENCES. 



209 



This theorem, like the preceding ones (16), (17), &c., re- 

 quires x not to transgress the limits x=l, x a 1. In the 

 case supposed (viz. when p and q are each equal to -J), (22) 

 becomes 



1 <c* ^i , r TT . rx f r \* 

 U XK - Z. * 1 cot - - sin TT COS-TT ......... (31). 



a a 2 a \ a J 



But as the party cannot be won in less than x games, 

 U XQ = u a while z is less than x, and therefore 



x ^' 1 + cot- ^ sin TT/COS- TT) ......... (32), 



a 2 a V a J 



of which (30) is a particular case. 



If, instead of seeking the probability that at the (z + l) th 

 game N would lose the party, by losing the last of his x 

 counters, we had sought that of his having at the termination 

 of this game any assigned number of counters &, the following 

 method might have been made use of. 



Let y xz be the probability in question. It is clear that it will 

 satisfy, as before, equations (1) and (2). But instead of (3), 

 we shall in this case have 



, unless x = k 1, and 



+ 2 (p -?)} (3). 



Equation (11) therefore, which depends merely on (1) and (2), 

 will still obtain; but instead of the system of equations (13), 

 we shall have the following: 



= C. sin - 

 a 



&c. = &c. 



. Ar-1 



sm TT + 



= &c. 

 = (7 t sin 

 &c.=&c. 

 = 



k+l 

 a 



. a-l 



sm TT 



a 



sm 



sm 



sin - TT + ... + 

 a 



. sn 





7T 



7T 



....(13'). 



14 



