208 Dr. L. N. G. Filon on a Sew Mode of 



^— denoting partial differentiation when a and D are inde- 



pendent variables, and D having the same meaning as before. 

 Hence, for any function F which can l)e expanded in powers 



DF(/(M)D)=g^^F(/(^)D) + F(/(«)D).D. 



dFn ___ —U , // / ^ d\ d ^ 



" df, " ^T-Ji^'^'W^-t^-T^Jdi:^ 



But Jo'=-Ji ; 



^^P"_,,p , f^ pi 



Now putting m = into (46) and remembering that P^=P„, 

 we have 



Hence, substituting for P\-i, 



dVn rP»-l f^P« "1 ,;.-,x 



From this 



-(n-l)[P,_,-;.P._i]. (52) 

 But from (50), putting ??2 = 0, 



(^l + l)P. + l + HP.-l-A<^» + l)Pn 



nP,+ {n-l)Vn-2 = IJi{^n-l)Vn-i. 



Substituting for P^+i and Pn_2 into (52), we obtain, after 

 reduction, 



^^ dir=^^''^'-^^' (^3) 



17. An interesting identity of two series of Legendre 

 associated coefficients, one proceeding in a series of varying 

 ?)i, the other in a series of varying 71, follows at once from the 

 expression of PJT as a Bessel operator on ijl'\. 



