198 Dr. L. N. G. Filon on a New Mode of 



Comparing (15) and (14:), 



when n is an integer, and it is 



= HT^ '^n^^~^".W^ • • • ^^^^ 



for other values of n. 



Lord Rayleigh ('Theory of Sound,' vol. ii. p. 230 of the 

 1st edition) has given a curious parallel to this, namely, 



giving the BesseFs function of order half an odd integer as 

 a Legendre coejQ&cient operator acting on an elementary 

 algebraic expression. 



8. From this symbolic expression for the Legendre func- 

 tions, it is easy to deduce that the Legendre coefficients 

 P„(/i) are the coefficients in the expansion of (1 — 2yLt^ + ^^)~2 

 in powers of t. For since, putting w = in the last result, 

 we have 



= Jo(v/I^2|^)(l + ^y^+... + ^>" + ...) 





i-^'.=o 



i'r- \ 



%/l-2fit + t'' 

 The expansion of — 7==^=^ can be shown to be abso- 

 lutely convergent if \ t \ < 1. And the expansion of 



