246 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 

 For V put S(r".P n ), then 



The coefficient of ?" in this equation becomes 

 (n+l)P H+l -2 



Equating this to zero we get 



(3). 



5. If we express the values of the Legendre's coefficients by means of 

 equation (2) we get 



= identically. 

 Hence P B+l -P w . 1 -(2n+ 1)^ = ........................... (4). 



Differentiating the left-hand side of (4) with respect to p. we get 



