OF ARTS AND SCIENCES. 143 



These equations, substituted in the first of the fundamental equations, 

 give 



f{i,j) = A\\ - h) [1 -f{hj, - I)] +f{JiJ - 1). 



Let C==iA"{l — b), 



and this equation gives 



1 -fi^.J) = (1 - c) [1 -Mj-1)] = (1 - cy 



F{i,j)=A^-{-{l-A')[i-(l-Cy-':\=l-{l-A')(l-Cy-' 



F(h, h) = i — {1 — A'^) (1 —oy'-\ 



which is the probability that A will win at the outset, if he has the 

 cue. 



PARTICULAR EXAMPLES. 



1. When the player A is an unfailing shot, we have 



a=l = A, B=h 



A''= 1 

 F(h, h) = 1. 



2. "When B is the unfailing shot, we have 



A = a, b=l = B, 



(7=0, F(/i,/i)=zA'' = a''; 



so that if (Ays average run is h, he can afford to give any player 

 whatever a grand discount, if he holds the cue. 



3. When A and B are equal players, we have 



a = b, A = B = -i-. 



If in this case the average run of each player is a little less than 27 

 points, or, more accurately, if it is 26.980, either player can venture to 

 give the other a grand discount, if he has the cue, and if the game is 

 50 points. 



4. Other examples for the game of 50 points are contained in the 

 following table : the first column contains the average run of the 

 player (A), who gives the gi-and discount, and the second column 

 the average run of the player (B). The chances of victory are equal, 



