LAPLACE. 549 



necessarily an integer, while m is. We may denote the value of 

 m by \xp + z, where z is some proper fraction, positive or negative ; 

 and then n = ^q — z. 



The r^^ term, counting onwards, in the expansion of {p-\-q)'^ 



after -^p^q- is ^ , p"-y^ 



\m \n 



w 



We shall now suppose that m and n are large numbers, and 

 transform the last expression by the aid of Stirling's Theorem ; 

 see Arts. 333, 962. We have 





m — r \/(27r) [ 12(/?i — r) 



^ = (?i + r)-"-'-^' e"-''- -7^ [1 - ^ 



• t • I t 



n + r ^ ' sl^'rr) \ 12 (?i + ?-) 



We shall transform the term {in — r)*"""^"^. Its logarithm is 



and wri-^U-'" '•' '' 



We shall suppose that r^ does not surpass //, in order of mag- 

 nitude, and we shall neo^lect fractions of the order - ; we shall 



thus neglect such a term as — ^, because vi is of the order //-. 

 Thus we have approximately 



^r - m - 2) jlog w + log ^1 - ^J I 



/ 1\ , r r^ r^ 



\^ 2/ ° , 2«i 2??i o»i 



and then, passing from the logarithms to the numbers, 



'-^ = m'--'"-^ 6*^2- ( 1 + -— _ — -J . 



(.. - r)— ^ = m-- e-2-^ 1 + ^^ - ^.) 



