248 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 

 7. Equation (3) gives 



Differentiating this TO times with respect to p we have 



(l m P 



1.1 . . 



m ~ l 



but from equation (4), 



1 n / \ ^ II 21-t-l 711 / r- \ 



therefore (2n+l) , w _, = - 7 m ;"~sr" (5)' 



or 



Subtracting (/S) from (a) we get 



ffm p Jin p Jm p 



''' 



r 

 8. Let ^ =( ,_^. 



) 



Then multiplying by (1 jr) 2 in equation (f>), we get 



or putting rt for n + l we get 



^^^'-^(^^-Oll-^fc 1 ........................ (7')- 



*-rrf :::-$ 



M 



Multiplying in equation (6) by (1 /^, 2 ) 2 , we get 



gL I ............... (8). 



n/r ij.- i L- a.- 1.2.3 ... (n TO) 



Multiplying this equation by ^ _ 3 _ 5 .. / 2M +i) ' we get 



( 9 ) 



