172 DE MOIVRE. 



Differentiate both sides with respect to t observing that 



Mh 



——-=:^aot. inns, 

 dt 



^ 2 { r - '-^^ aU + ' ^' -f ^^ - ^^ (abtr -... 



Now put r = Z + 3 ; and we obtain the required result. 

 Thus a linear relation can be obtained between the coefficients 

 of successive powers of t. 



This is De Moivre's first law ; see his page 205. 



1 _l_ a/(1 — 4c) 

 Again ; let iV= ^ ~ ; then the original expression 



becomes 



'\Jni+n /-| m+n 7vr-2wt-2n\ 



= d'fN-'' (1 - c'"iV^"'"0 (1 - c"*^"^-'"^-'")"'- 

 We may now proceed as in the latter part of Art. 304, to de- 

 termine the coefficient of r"*"^"". 



The result will coincide with De Moivre's second law ; see his 

 page 207. 



307. Problem LXV. is a particular case of the problem of 

 Duration of Play ; m is now supposed infinite : in other words 

 A has unlimited cajntal and we require his chance of ruining B in 

 an assigned number of games. 



De Moivre solves this problem in two ways. We will here 

 give his first solution with the first of the two examples which ac- 

 company it. 



Solution. 



Supposing n to be the number of Stakes which A is to win of B^ 

 and n + d the number of Games ; let on- 6 be raised to the Power whose 

 Index is 7i + d; then if d be an odd number, take so many Terms of 



that Power as there are Units in — ^ — j take also so many of the 



Terms next following as have been taken already, but prefix to them 

 in an inverted oi^der, the Coefficients of the preceding Terms. But if 

 d be an even number, take so many Terms of the said Power as there 



