210 ON THE SOLUTION OF EQUATIONS 



From whence, by the same system of factors as before, we 

 deduce the general formula 



4/2 fc+1 \* . kr r ,-* 



r = - ^j-i I sm TT cos - TT ............ (14 ) ; 



a \/r / a a 



for the factors corresponding to the two equations whose first 



(k 1) r 



members are different from zero, are sin - - TT and 



a 



. (k + 1) r , , , . . kr r 



sm f - ' TT, and the sum of these is 2 sin TT cos - TT. 

 a a a 



Consequently the expression of the probability sought will be 

 (accenting the y for distinctness), 



x-k 



2 . ^.fp\ 2 ~ n -i . kr . rx / r ^ ,.._.. 

 y' =- Upn) 2 ] 5 'sm TT sm TT cos- TT ... (15). 

 a^ \qj a a \ a J 



It is an obvious consequence of the discontinuity of the limiting 

 conditions of the problem, that this expression does not reduce 

 itself to (15) when k is taken equal to zero. For the same reason 

 it is not applicable when k is equal to unity : and on the other 

 hand, it is not to be greater than a 2. 



It is unnecessary to trace the different corollaries deducible 

 from the last written equation, as it has been introduced merely 

 to illustrate the facility with which our method discusses any 

 proposed modification of the question of the duration of play. 



One point, which is perhaps worth notice, is the symmetri- 

 cal manner in which x and ^?, k and q, enter into (15') : the 

 result, however, which is the interpretation of this symmetry 

 may probably be obtained by general considerations. 



A more general question would arise from supposing it 

 possible for M to win or lose at each game any number of 

 counters not greater than a. The method we have been 

 illustrating would apply to this question, but the solution of it 

 involves that of an algebraical equation of a degree superior to 

 the second. 



Another part of the subject, namely, the numerical calculation 

 of the expressions already obtained, would not be consistent with 

 the design of this paper. When z is sufficiently large, all the 

 summations with respect to r may be reduced to their first and 



