CONNECTED WITH GAMES OF CHANCE. 177 



11 n 



In this case n 1 = 1, n s = — ^,n, = — g, w 4 _ n, 2: u 



and we have 



W = « 2 P . 



o? 









P 



— 



q+T 





2 r 



— M 





9. 



M 







u . 



3 r ' 







3~ r 









The questions examined in the preceding pages afford an 

 instance of the immediate application of some very abstract 

 propositions of analysis to a subject of constant occurrence, 

 which being as far as 1 have been able to discover, hitherto 

 untouched, and also requiring reasoning of rather an unusual 

 nature, I have preferred treating in particular instances, in- 

 stead of investigating it in its most general form. The prin- 

 ciples on which similar problems should be attempted, are 

 first by means of some combinations of the roots of unity to 

 represent the stake after the decision of any number of events, 

 and then by means of any known theorems respecting the 

 roots of equations, to ascertain the sums of the series which 

 present themselves in the result. 



VOL. ix. p. 1. z XI. 



