MOXTMORT. 119 



1.2 s^ (l-At/"' 



1.2.3 6'^ (l-/x) 



\-2 



-^^!i rli I til ^^^-^ tn{i-i)(i-2) ■ 



- ,,,U » i + ^^+ ^^2 + 1.2.3 "^••• 



71^-^^ 



where the series between square brackets is to extend to X + 1 

 terms. 



We may observe that by the nature of the problem we have 



a + Z> + c + ...=0, and also z+y + x+... = 0. 



The problem simplifies very much if we may regard I as infinite 

 or very great. For then let z denote the advantage of -4 ; if ^ ob- 

 tains the next deal we may consider that his advantage is still z ; if 

 A loses the next deal his advantage is the same as that of B 



originally. Thus 



mz + n2/ 



s 



MultijDly by s and transpose ; therefore 



z = 7/-{- aq. 

 Similarly we have 



y = x + hq, x = ii + cq, 



Hence we shall obtain 



^ = 2L(^-l) + &(p-2) + c(i?-3) + ...j, 



where p denotes the number of players ; and the values of y, ^, . . . 

 may be obtained by symmetrical changes in the letters. 

 We may also express the result thus, 



z 



= _£|a+2^>+3c+...|. 



