170 . DE MOIVRE. 



We can tlius obtain by the ordinary doctrine of Series, a linear 

 relation between the coefficient of f^^"" and the coefficients of the 

 preceding powers of t, namely, r'^^^ ^""^^"^ ... This is De 

 Moivre's first law; see his page 198. 



Again ; we may wiite the above fraction in the form 



JV" (1 + c"iY"^") ' 



, 1 + ^(1 _ 4^ahe) 

 where N = --^ ; 



2 



and then by expanding, we obtain 



The coefficient of f'' in N'" is known to be 



^j^n {n-{-.x-\-l) {n -{-x + 2) ... {n ■\- 2x — 1) ^ 



' X ' 



see Differential Calculus, Chapter ix. 



Similarly we get the coefficient of f^'' in N-'\ of i'^'*" in 

 iV"^**, and so on. 



Thus we obtain the coefficient of f'"^^ in the expansion of the 

 original expression. 



This is De Moivre's second law ; see his page 199. 



805. De Moivre's Problems LX. LXi. LXii. are simple ex- 

 amples formed on Problems LVIII. and Lix. They are thus 

 enunciated : 



LX. Supposing A and B to play together till such time as four 

 Stakes are won or lost on either side ; what must be their proportion 

 of Skill, otherwise what must be their proportion of Chances for win- 

 ning any one Game assigned, to make it as probable that the Play will 

 be ended in four Games as not? 



LXI. Supposing that A and B play till such time as four Stakes 

 are won or lost : What must be their proportion of Skill to make it a 

 Wager of three to one, that the Play will be ended in four Games % 



LXII. Su2:>posing that A and B play till such time as four Stakes 

 are won or lost ; What must be their proportion of Skill to make it an 

 equal Wager that the Play will be ended in six Games ? 



