DE MOIVEE. 181 



This is very easily demonstrated. For suppose that one scale 

 of relation is (1 — of ; then by forming the ]3roduct of the cor- 

 responding terms of the two Recurring Series, we obtain for the 

 general term 



=== a" {Rj>: + E,p: + B,p: +...] 



\n 



Tliis shews that the general term will be the coefficient of 

 r" in the expansion of 



(l-rapy {1-rap;)' (1 - rap^y '" ' 



and by bringing these fractions to a common denominator, we 

 obtain De Moivre's result. 



822. De Moivre applies his theory of Recurring Series to 

 demonstrate his results relating to the Duration of Play, as we 

 have already intimated in Art. 313 ; and to illustrate still further 

 the use of the theory he takes two other problems respecting j)lay. 

 These problems are thus enunciated : 



Lxx. M and N, whose proportion of Chances to win one Game 

 are respectively as a to h, resolve to play together till one or the other 

 has lost 4 Stakes: two Standers by, j5 and S, concern themselves in the 

 Play, R takes the side of M, and S of N, and agree betwixt them, that R 

 shall set to S, the sum L to the sum G on the first Game, 2L to '2G on 

 the second, 3Z to ?>G on the third, 4Z to AG on the fourth, and in case 

 the Play be not then concluded, 5L to 5G on the fifth, and so increasing 

 perpetually in Arithmetic Progression the Sums which they are to set 

 to one another, as long as M and iV play; yet with this farther con- 

 dition, that the Sums, set down by them R and aS', shall at the end of 

 each Game be taken up by the Winner, and not left upon the Table to 

 be taken up at once upon the Conclusion of the Play: it is demanded 

 how the Gain of R is to be estimated before the Play begins. 



Lxxi. If M and iV, whose number of Chances to win one Game 

 are respectively as a to h, play together till four Stakes are won or lost 

 on either side ; and that at the same time, R and S whose number of 

 Chances to win one Game are respectively as c to d, play also together 

 till five Stakes are won or lost on either side ; what is the Probability 

 that the Play between M and iV will be ended in fewer Games, than the 

 Play between R and S. 



