204 Dr. L. N. G. Filou on a Ke.w Mode of 



Repeating this m times, we find 



_ '" rp"- 



{l-;.-^)-.P;;'(^) =--?,(;.). .... (3.5) 



a well-known relation. 



13. I£ we integrate the right-hand side of the equation 



n times with regard to iju, and equate Pn(At) to the nth. differ- 

 ential coefficient of this integrated right-hand side, we obtain: 



l—^\n 1 Jn I 2n-l 



(_1V^ f/« 1 i 2«-l 



= -1.3...(2»-l) a^.^'^-^"'' f^,-]n-'-^a--')^du, (37) 



where u=(t — /ji)l V fju^ — l^ and the integral is taken round a 

 contour in the 2^-plane inclosing the points ?^= +1. 

 Thus 



Hence by the previous result 



7)1 



p^i^)=%^-'a^('^'-^y'- ■ ■ ■ m 



So that all the well-known forms for the Legendre coeffi- 

 cients flow^ quite easily from their definition as multiples of 



14. Further, it can be easily shown that the recurrence 

 formulse for the Legendre functions follow directly from the 

 recurrence formulae and the differential equation for Bessel 

 functions. And in this it will not be necessary to suppose 

 that n is an integer, but we may suppose it anything we 

 please. 



