206 Dr. L. N. G. Filon on a JVeic Mode of 



Substitutiuff this latter value of J \ we have 



W) 



^'p»'=/.p;;,Li-v/r=:^p»'_^\ . . . (46) 



another recurrence formula. 



Eliminating P^ between (44) and (46), I find 



/J-— 2P„_i= (?i — ?»)^»-i + (?i + ?>0-^H-i ; 



replacino- the P's by the P^s and writing ?z + l for /i, this 

 gives 



(,,_,,,+i)(,,+„,)P--i+P-+i^ -==^p;r, . (47) 



v/i— M" 



a well-known recurrence form. 



15. The recurrence formula in which m is constant and n 

 varies by an integer may be similarly obtained. 



But the BessePs functions satisfy the relation 



J..."(z) = -l J„:{z) + d' - 1 ) J».(-) . . (48) 



