268 Mr. Nicholson on Legendre's Functions and the 



" ,. f -== M = 2 f ? «* 



° Vco S ^ -cos** Jo -^/l-sin^sin'A 



= 2K(sinf). 

 Hence 



K(co S |) = K( s in^) = 2 2^ i P„ W . . (9) 

 It follows from (8) that 



■£p,WK( v /I^)*- 3 A_ i . . (10) 



or 



J/KP„(2F-l)^=^ (ID 



Now when n is large, 



P-M=\/— * 5 - cos {(^ + 1)0+^1 . (12) 



V ft7T Sill I 4 J 



and the series (8) and (9) are accordingly absolutely con- 

 vergent between = and ir, but at these limits the theorems 

 are not true. The integrals further deduced, however, may- 

 be taken over the entire range, the integrals being finite at 

 each limit. 



Squaring (8) and integrating with respect to //., 



,{lx)djL 

 1) 



1 



P KWm-4 i P P - 2 >-^ 4-4 5 P - P -M p -M d 



J-i C } ^~ oJ-i (2»+l) a + -J-i(2n + l)(2m + 



= 8 

 Let 



o (2n+l) 3 " 



111 



^Smi+i — p TO+ i + 2im+T+ 32m+l + \ 16 ) 



f 



ifcKWrsjcr, (14) 



The formulae arising in a similar way from (9) are 



fp-W-K-fv^-H, ' . (15, 



j; 



i) : 



kK'P„(2F—l)d!c = ^'^ . . . (16) 



