DE MOIVRE. 171 



806. Problems LXiii. and LXiv. amount to the general enun- 

 ciation we have given in Art. 303 ; so that the restriction that 

 m and n are equal which was imposed in Problems LVIII. and 

 Lix. is now removed. As before De Moivre states, tuithout de- 

 monstration, two general laws, which we will now give. 



Laplace shews, T]ieorie...des Prob. page 228, that the chance 

 of A for winning precisely at the {n + 2u?)*'^ game is the coefficient 

 of f^'^'^ in the expansion of 



f 1 + V(l - 4c) ) " f 1 - V(l - 4c) 1 "^ 



r-{ 



m 



1 9 9 



j 1 + ^(1 _ 4c) r"^'^ ( 1 - v(i - 4c) I '"^'* • 



Let s) — ^^ denoted by h \ then the fractional expression 



which multiplies a"^*" becomes by expansion, and striking out 2h 

 from numerator and denominator, 



ay^-' m{m-V){m-2) /1\'""%2 w(772-l)(w-2)(m-3)(m-4) [ly^, 



. ^lY"--^"-^ , (m+^2)(m+^^-l)(m+?i-2) /l^""^"-",, , 

 (^+^)(2J + g %) ^^+- 



We have to arrange the denominator according to powers of 

 t, and to shew that it is equal to 



where 7 = m + n — 2. 



Now, as in Art. 30-A, we have 



{ 



1 + V(l - 4c) I" ^ 1 1 - V (l - 4c) r 



= l_,e + '-(p^8) ^,_ r(,-4)(r-o) ^3_^ __. 

 and the left-hand member is equivalent to 



