378 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



(n m l) (n m 2)(n + m + 3)(n + m + 2) (n + m+l) (n + m) 



(2n-l)(2n+lY~ 



rO 1 



.. . u - 



(n + m + 3)(n + m + 2) ... (n + m -2) ,,, 

 (2n-8)(2n-l)(2n+l) 



Hence l- 



3)(n m + 2)(n m+l) (n m) (n m 1) (n HZ. 2) ,-,, p 

 ) (2 + 3) (2n + 5) ~ " 



o (n-m+l)(n-m)(n-m-l)(n- m - 2) (n + m + 3) (ft + m + 2) n , n p 

 (2n-l)(2n+l)(2n+5) 



(n-m-l)(n m- 2)_(ft_+m + 3) (n + m + 2) (n + m + l) (n + m) 



~(2n ^3)(2n+l)(2w + 3) "- 1 



(n + m + 3)(M + m + 2)(n + m+ 1) (n + m) (n + m 1 ) (n + m 2) _ w 



2n-3~2'/ ~ ~ " - 3 ' 



The law which is here observed is also found to hold for (l ju' 2 ) 4 J D m+4 P, 1 

 and is true generally. 



The general term of (l-p}"D m+p P n is 



ff( f) -l)(j?-2)...(j?-r+l) 



r! 



2r) ... (w m _p 



x 



This may be expressed under the form 



(-IV"* -^ ! (n-m+p-2r)\ (n + m+p)\ 



r I (p r) I (n m p) I (n + m +p 2r) ! 



. V. V. g . p,! - 2 ". 



As a test of the correctness of this result, when m = Q, this reduces 

 to the expression previously found for (l fjf) p D' > P n . 



