254 USEFUL FORMULAE, CONNECTING LEGENDRE'S COEFFICIENTS, WHICH 



hence 



{(, -,= 



where 



14. Since 

 we have 



- . 

 = (! /x") 2 -r-^cos (m\ + a). 



- 



Combining this with equation (10), we get 



a very simple formula for expressing - 





in terms of Q'" +1 and 



(13), 



15. Multiplying equation (4) by n and adding to equation (3), we get 

 (2n+l)P n+l =(2n+l)pP n +n(2n+l)!p n dp, 



or 



Multiplying equation (4) by 

 we get, similarly, 



Hence differentiating we get 



and subtracting from equation (3), 



and 



dp, 



therefore 



(14). 



