ULTIMATE RESULTS OF PLAY. Ill 



same equation (1) is equally true if forB W}00) we substitute 

 either 1 B W)00j or AOO, n> which two last may be 

 different, for any thing yet proved to the contrary. In 

 fact, the equation (1) is merely the general expression 

 of the condition that n is changed into n -j- 1 or n 1 

 according as one or the other of two events happens, 



b a 



whose chances are and It may be seen 



a -f b a + b. 



however, immediately, that in the case of n = 0, in 

 which case the proposed contingency becomes an initial 

 impossibility, we must have B 0> , = 0, or C' + C" =0. 

 We have then 



= "{-(9"} 



This result is rational only when a is not greater 

 than by unless we suppose C" = 0. But the necessity 

 for investigating what takes place in this case is saved 

 by observing a very simple relation which exists between 

 B w> w> and B w>00 . Supposing A to have infinite means, 

 it makes no difference in the state of the question if we 

 take any number of stakes m from the stock of A, and 

 suppose that they shall be lost before the rest are touched. 

 Consequently B cannot win indefinitely from A unless 

 he first ruin A's stock of m stakes, and afterwards, be- 

 ginning from n + m stakes, win indefinitely from A's 

 remainder. That is 



m,oo 



-, n n \ 



b a ) 



Whence, applying the same reasoning to A m> Wf we 

 find that if two players A and B, possessing m and n 

 x 3 



