274 Mr. Nicholson on Lecjendre's Functions and the 



Now 



/ 1 2 IV- ^ 



\2n-l 2n + l + 2^ + 3/ ~ {2n-l)\2n + l) 2 (2n + 3) 2 ' 



*'' ( l (W(k))*kdk 



128 2n + l\2w-l ' 2w + 3 2n + l/ 



"128 o L2n+l\(2n-l)2 + (2»+3) 2+ (2n+l) 2 / + 2J 



~32 o(2rc + l) 3 ' 

 and if o- 3 =p + 23+ 33 + 



'*• (\w(k)Ykdk = ^cr, (38) 



We now integrate (35) by parts, 



(-) }l - ] 2 r-E'(*) dP n y 



(2w-l)(2w + 3)(2n + l) 2 Ln{n + 1) K ^ ] dp J _! 

 1 f l <?P„ rfE'(fc) . 



•. f l rfP,^ ,, (-) - 1 2n(n+l ) 



J ^^ dfi dk afC ~ (2n-l){2n + Z)(2n+l)*' 



or 



rA 3 (E'-K')P„'(2A 8 --l)dA 

 Jo 



( — )» n(n + l) 



~~ 2 '(2n-l) (2?i + 3)(2n + l) 2 

 Changing the variable from k to #', 



f 1 A(l-A: 2 )(E-K)P n / (2F-l)^ 



_ 1 n(rc + l) 



~2 ' (2w-l)(2n + 3)(2w + l) 2, 



(39) 



(40) 



Combining (40) with (30), and (39) with (31) respectively, 

 we find 



\\*EP' n (2k*-l)dk= ^ n ^]l { 1- ,,, 9 . * « } 



Jo (2n+iy I 2{2n-rd)(2n — \) ) 



n(n+l)(Sn 2 + Sn-7) 

 ~ 2(2n + 3)(2n-l)(2n + iy> ' (41) 



