374 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



A (r) A (n - r) A(q- r) 2ro + 2g-4r+l 

 ~ ~ 2q-2r+l' 



... 1.3.5 ...(2r-l) 

 where A(r) = Ii2 . 3 ...r "' 



as in a previous paper (see Vol. I. p. 492). 



Also if p = q, the coefficient of the general term reduces to 



1)- 



Also if p is greater than </, or if m is greater than n, the whole 



expression vanishes. This seems to imply that 7 ,-. and - - occur in 



(qp)- (n m)l. 



every term. 



If m + p is greater than n + q, the values of the terms become in- 

 determinate, since whatever their coefficients may be, they will be made 

 to disappear by differentiation. 



This would seem to imply that (n + q m p) ! is a factor. 



. . , 1.3.5 ...(2r-l) 2r\ 

 The expression A (r) = ^- = ^ . 



Hence 2 r (rty- A (r) = 2rl. 



d m P 

 2. Let us now express , " or D m P n in terms of P n _ TO _. r , adopting 



/ d \ m 

 the notation D m for I -? 1 as an operator. We have 



Hence by successive additions we get 



DP n = (2n - 1) P n _! + (2n - 5) P n _ 3 + &c. + 3P, if n be. even, 

 or DP n = (2n-l) P n _, + (2n - 5) P n _ 3 + &c. + P if n be odd. 



By differentiating and substituting for the first differential coefficients 

 )l = (2n-l){(2n-3)P B . 2 + (2n-. 7)P n _ 4 + &c. +P } 

 + (2n - 5) {(2n - 7) P._, + (2w - 1 1) P n _ 6 + &c. + P } 

 + &c ....................................................... 



+ 3P , if n be even. 



