276 Mr. Nicholson on Legendre's Functions and the 



(1 1 * ( — Y l 



2 o (2w — l) 2 (2n+l) 8 (2n + 3) 5 



=±i (-)" 



32 o(2n+l) ; 



7T° 



~I024* 

 Hence 



\ KEWdk= -^-r = i f &KKWL . . (49) 

 Jo !024 64 J 



Again, by (43) and (9), 



f\^ 



£EK'<flfc=-2 



J o(2* + l) 2 (2n+l)(2n-l)(2n + 3) 



= % (2»-l)(2n + 3)(2n+l)' 3 ' 



Jo o (2w — 1) (2n + 3) (2n + l) d 



= ( 1 AiEK / ^ (50) 



Now ek'+e'k=kk'+| 



(Cayley, Elliptic Functions, p. 48) 

 ( X kEK'dk= if'ib -TkK' + g"Vd* 



_ 1 /7T 8 7T\ 



~2Vl6 + I/ ? 



r*EK^=f 1 ffl , K^'=^ + ^. . . (51) 

 Jo Jo ^ ° 



Again, 



Jo ^ KE ^' = - S o (2n-l)^n + 3K2n + iy»' 



aUd f l °° 1 



Jo * K ' E '^- ~ S o (2»_1) (2* + 3) (2* + l)» ; 



r*KEdife= pfcK'E'dfc .... (52) 



