DE MOIVRE. 169 



,10 



(a + by 



De Moivre leaves his readers to convince themselves of the 

 accuracy of his rule ; and this is not difficult. 



De Moivre suggests that the work of multiplication may be 

 abbreviated by omitting the a and h, and restoring them at the 

 end ; this is what we now call the method of detached coefficients. 



304. The terms which are rejected in the process of the 

 preceding Article will furnish an expression for the probability 

 that the play ivill be ended in an assigned number of games. 

 Thus if ?i = 4 and d = ^, this probability will be found to be 



a' + b' ^a'b + ^ab' l^a%'' -\-\^a%' 48a^6^ + 48a^^>^ 

 {a-^by^ {a + bf "^ {a + hf "^ . {a-\-bf' * 



Now here we arrive at one of De Moi\T:e's important results ; 

 he gives, luithout demonstj^ation, general formulae for determining 

 those numerical coefficients which in the above example have the 

 values 4, 14, 48. De Moivre's formulae amount to two laws, one 

 connecting each coefficient with its predecessors, and one giving 

 the value of each coefficient separately. We can make the laws 

 most intelligible by demonstrating them. We start from a result 

 given by Laplace. He shews, Theorie . . . des Prob., page 229, 

 that the chance of A for winning precisely at the (n + 2x)*'* game 

 is the coefficient of T"^^ in the expansion of 



( i + ^{i--^abf) r M - V(l - -^abt') r ' 



I 2 l^-j 2 J 



where it is supposed that a + b = 1. 



Now the denominator of the above expression is known to be 

 equal to 



^ 1.2 [3 -t ... 



where c = abt^ ; see Differential Calculus, Chapter ix. 



