272 Mr. Nicholson on Legendre's Functions and the 

 giving 



( kKV n (2k 2 -l)dk 

 Jo 



1 

 which is equation (10) again. 



We now proceed to treat (15) in the same way as (10) . 

 The formulae connecting E' and K' are 



S' = -i (E '- K ') • • • • (32 a) 

 dK 



dk kk 



(15) gives 



'--£>{*-**■'}. • • W 



(Zn+ly Ln{n + 1) 'dpj-i J_i?i(« + 1) d/j, tip J ^ 



Jo v y dju- dk v y (2?i+l) J 



or, by (32 b) , 



J^{PK'-E'}P,/(2^-l)^=(-)» g-J-yj*. 

 a repetition of (31) by a different method. 



We now consider another expansion in a series of Legendre 

 functions, commencing with the equation 



4 fcos^r cos&sr cos 5z ( — )"~ 1 cos (2n + l)z ~] 



1-f cos 2z =-H -Y-gr + t r -g- , ~ — ^ ~ 7 ...+ /o~_iw^ i iwo„ . o\ + -.« 



(33 



ar L 1-3 1.3.5 3.5.7 •"• + (2n-l)(2w + l)(2w + 3) + "'/ 



It leads to 



1 f* (1 + cos 0)# 1 J» ( -)*_^P.>) 



2V2jo y/co S <j>-cos0 3 "W* i(2»-l)(2n + l)(2»i+3) 



But 



l + cos<£ l + cos0 



— /- — ~ ==* ~ V cos 6- cos 6 -i -. -r- - , 



v cos 0— cosy VCOSCp — COS0 



J.. x /«4-« rf i 



\ a/ cos 2 * - cos- 5 # = 2 1 — — 

 ~° N 2 2 Jo (l + 



with the usual substitution. 



tan ^ . sin 2 \dX 

 1 1 + tan 2 ^ cos 2 X V" 



