282 Mr. Nicholson on Legendre's Functions and the 



By changing to it — 6, 



K(cos-i*- f) - } [ K'(co S f ) # = 2f (^P»W. (08) 

 These with (8) and (9) give the curious results 



\W+Tf> • • ((>y) 





7T if 



to > 2 



P.(cos 0) 



0\ r i o 2n + l 



= 4 f (-)»P»(cOS0) 



o (2*-fl) 2 ' 1<UJ 



7T »)7T 



which hold from = to 2tt except at ^ , — } . 



If x rKL x (*(-*)* 



J o K ( cos 2 j j. K ( C0S 2) 



We may at once deduce certain integrals as follows : 

 A'(47r-2cos- 1 / L - + I 1 ).K'.^=2 (71) 



£(l 2 -7r+2cos- 1 £).K.(/*Z:=2 (72) 



( 1 ^(47T-2cos- 1 /.fT ] ).K / .P w (2F-l).^=^ 3 . (73) 

 Jo 4 



| ^(I 2 -7r + 2cos- 1 A0.K.P n (2/.- 2 -l).^-=f . . (71) 



(^(477-2 cos" 1 k + I ] yK' 2 dk=^a- (75) 



Jo * 



(^(I 2 -7r + 2cos-H") 2 .K 2 .^- ^<7 5 . ...... (76) 



j: 

 j: 



