OF THE FORM d m P n /d^ m x dPP^dp?. 375 



The coefficient of P is 



(2w-l) + (2w-5) + &c. + 3 to terms = n ( n+l \ 



2t 2t 



The coefficient of P 2 is 



r,,i ~> . r- n 9 (re + 5) (re 4) 



I he coefficient of P 4 is - ' v - , and 



so on. 



The coefficient of P B _, is (2re-3) %~ = (2n- 3) (2n- 1). 



Zs 



Hence when n is even we have 



+ 2(r+l)(2n-2r-l)(2n-4r-3)P )l _ 2 _ 2r + &c. to 



Similarly when n is odd we have 



= 2 (274 - 1) (2n - 3) P,,_ a + &c. + 2 (r + 1) (2n - 2r - 1) (2n - 4r - 3)P, 1 _ 2 _ 2r 



+ &c. to (n-l)(w + 2)3P 1 . 



Following the same method of expansion we get values for the suc- 

 cessive differential coefficients, 



n = 2 {(r + 1) (2n - 2r - 1) (2n - 4r - 3) 



+ 4 (2n - 3) (2n - 7) {(2n - 9) P n _ 6 + &c. + P } + &c. 



+ 2(r+l)(2n-2r-l)(2n-4r-3){(2n-4r-5)P B _ sr _, + &c. +P } 



+ &c. + (%-!)( + 2) 3P , when is odd. 



The coefficients of the successive terms when collected give the law 

 of formation as follows : 



The coefficient of (2n 4r 1) P n _ 2r _, is 



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



and the last term is that when n 2r 1 = or 1, so that the coefficient 



,, D . (n-l)(n+l)(n + 2)n , 

 of r", is - ^^r ^- - when n is odd, and the coefficient of Pj is 



, 

 when re is even. 



LJ . 



