270 Mr. Nicholson on Legendre's Functions and the 

 If m is equal to 5 or less than n, 



,. =0 Am+n-r \2n + 2m — 2r+lJ n +m-2r 



where ^ T 



Ar = 2^ip (24) 



Now 



j-i K> ( cos i) p » {ix)dfx = s ?o^r r k ( c ° s i)p^)p.M^. 



Write (23) in the form 



F m (z)F n (z) = 2 (\r) P (*)... (25) 

 where mLn\ 



= S o 4 v f ' K ( cos f ) s * M p ( cos ^ 



S=0 2.5 + 1J_1 \ 2/ r= o\ A«+ S -2r 



+ 2 ^-4t\ K(cos*)2(X r )P (cos0)d/a 



s=m+1 AS-f- 1J— 1 V 4/r=0\ ' s n+s~2r 



= 2 2 - 2 -Yx,Y - 



s =o r=o 25 -f 1 V ' A" 2>2 + 2s — Ar + 1 



«p J 1 2 / \" 2 



+ . J +1 r ? 2^+1 V r A 2n + 2*-4r+L" 



•• f^p^-i)^£f [2t+1)( ^ 8 _ 4f+1) 



+ 2 X r)s . . (26) 



.=t+irto(2* + l)(2n + 2«-ir + l) V J 



and 



•- f 



Jo 





&KM = 2222 



»=o*=o r=o {2s + 1) (2n + 1) (2w + 2s— 4?- + 1) 

 + 22 2 2 [ ±li (27) 



n=0 *=»+l r=0 (25 + 1) (2n + 1) (2ra + 2* - 4r + 1) v ; 

 where (Xr)* = (X,.)*, and is defined in (25). 



