392 ON THE PRODUCT OF ANY TWO LAPLACE'S COEFFICIENTS 



12. Putting r 1 for r in the general term of the expression for 

 Or-l)" D m P n xD"P p , we have 



2pl (2n-2r + 2)! (n + m)\ (n-m + 2p-2r + 2)l(n+p-r+l)\ 

 ' (r-l)l(p-r+l)l(n-r+l)\(n-m)!(n + m-2r + 2)!(2n + 2p-2r + 3)\ 



x (2n + 2 P - 4r+5) IT->P n+p _ f+t . 

 Multiply by (2p + l) and we get 



> 



x (2n + 2p 4r- 

 Also writing down two terms of (ft 2 l) p D m P n x D p P p+t multiplied by 



we have 



y-i - -2r + 3)\(n+p- r + 2)_!_ 



(r - l)\(p - r + 2)1 (n -r+ 1)1 (- m)l(n + m^2r+~3)\ (2n + 2p -2r + 5) I 



x [(2w - 2r + 3) (2p - 

 x (2p + 3){(n-m + 2p- 2r + 4) D"^P n+ii _ 2r+t + ( 



, _,y (2p+l)l(2n-2r)\(n + m)l(n-m + 2p-2r+l)l(n+p-r+l)\ 

 ' -ln r\-m 



x [(2i - 2r + 1) (2p - 2r + 2) - 2 (p + 1) (n - ni)] 



x (2p + 3) {(^ - m + 2p - 2r + 2) D m ~" P,^,^ + (n + m - 2r + 1 ) D m -P n+p ^}. 



Now take the coefficient of D m ~* P M+p _ 2r+2 in the sum of these terms 

 and we get the corresponding term in 2(fji'l) I> D" l P n xD''P ll+z 



2)\(n+p-r 



r ! (p -r + 2) \(n -r)l(n- m)l (n + m-2r + 2)l (2n + 2p-2r + 5)l 

 x 2{f2p+S)(p-r+2)(n+m-2r+2)(2n+2p-2r+5) 



x [(ft + m - 2r + 1) (2p + 2)-2r (2n - 2r + 1)] 



- (2ft -2r+l)(n-m + 2p-2r + 3) r [(2ft - 2r + 3) (2p - 2r + 4) - (2p + 2) (ft - m)] 

 2(2n2r+l)r(p-r+2)(2n+2p-2r+5)(2n+2p-4r+5)}, 



