202 Dr. L. N. G. Filon on a yew Mode 0/ 



Suppose I i I large, so that | « | is large. 



Then we may, if \u\ be taken sufficiently small, expand 



1 . „ w , 1 . .1 



m powers or - and — — r^^ ijowers 01 — 



u—u ^ a J ^ ya 



Mence u 



1 _ /'""* '*'" '""* ( — 1)'"\ 2a cosec 6 , . 



(27) 

 i— COS 6- 



Now 



14-^2 = 2^ coseo e^/t^-'lt COS ^+ 1. . . . (28) 

 So that finally 



t—cose 



_ ■^^(,_ Ij y^-^-2^ COS ^+ 1 L J, «« + i, -^i^nr] 



The coefficient of ?i'" is 



1 sin'" e 



^t^-2t cos 6-\- 1 (i-cos 6>+ y^2_2^-(jQs 0^iy 

 = ( — 1)"^ coeff. of li-'". 



Hence '- — , Vl^ia) is the coefficient of r""! in the 



expansion of 



1 (1-/^^)2 - 



(29) 



in inverse powers of t. 



When m = this gives the Legendre's coefficients as the 

 ceefficients in the expansion of {t^ — 2t/jL+l)~^, as it should. 



12. In the above t may be anything. Let us take t a com- 

 plex variable. Then remembering the expansion is one about 

 infinity, we have for the coefficient of t~'^~^ 



where the integral is taken round a circle (or any contour) 

 inclosing all the singularities of the integrand. Here the 



