394 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



Or substituting 2q for (2p + 4) and q+2 J for (2p + 2) we get : the coefficient of 



in the expansion of 2 (/**- I) 1 ' D m P n x D p P q 



/ _ , \, (<Z + p) ' (2n 2r) ! (n + m)l (n m + q+p 2r) ! (n + q r) ! 



' r\ (q-r)l (n-r}\ (n-m}\ (n + m-2r + q -p)l (2n + 2q-2r+l)\ 



(q-r)\ (n-r)l (n-m)l (n + m-2r + q-p)l (2n + 2q 



'2q (2q-l) (n + m - 2r + l)(n + m-2r + 2) 



-2(2q-l)(n + m-2r+2) 2r (2n-2r+l) 

 + 2r(2r-l)(2n-2r+l)(2n-2r + 2) 



NOTE. It is readily shewn that the expression in this large bracket is 

 equivalent to 



[(2n-2r+2)(2n-2r+l) (2q - 2r) (2q-2r-l) 



-2(2n- 2r + 1) (2q - 2r) (2q-l) (n - m) 

 + 2q(2q-l) (n - m) (n -m-l )], 

 and also that it 



'(n + m - 2r + 2) (n + m -2r+l)(2q- 2r) (2q - 2r - 1 )' 

 - 2 (n + m - 2r + 2) (2q - 2r) 2r (n - m) 

 + 2r (2r - 1) (n -m)(n-m-l) 

 This expression is also 



'(n + m 2r + 2} (n + m 2r + I ) (n m + 2q- 2r) (n m + 2q 2r 

 -2(n + m-2r + 2) (n - m + 2q - 2r - 1) (n - m) (n + m + 1 ) 

 + (n m) (n m 1) (n + m + 1) (n + m + 2) 

 Substituting this last expression in the coefficient of 



(2n+2q-4r+l)D m -''P n+y . lr 

 in the expansion of 2 (/u, 2 - l)"D' n P n x D p P q , 



we 



r\ (q r)\ (n r}\ (n-m}\ (2n + 2q 2r+ 1)1 (n 



-2r + 2)(n + m-2r+l)(n-m + 2q-2r-l)(n- m + 2q- 2r) 

 -2(n + m-2r + 2) (n - m + 2q-2r-l)(n- m} (n + m+l) 

 + (n m) (n m 1 ) (n + m + 1) (n + m + 2) 



