120 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



After reduction I find, for s = 0, 2, 

 /,' (cos) = 4rf cos /3 cos y _ 



o SHI 



^ (4 _ 



Now 



9* = 2 - 5/c'V 3 2 (/c - *' 2 ) (1 - *V 2 ) ! , 

 27i fi = (4 - 7 *V 2 ) (/c- - K"~) (4 - 13/cV 2 ) (1 - /c/c' 2 )*, 

 81 1* = 8 - 40/<V 2 + 4 1 /cV 1 4 (2 - 5/cV 2 ) (/c 3 - /c /2 ) (1 - ^/c' 2 ) 1 . 



Whence on substitution, with t /" 2 ,(/ 3 = 1 , 



47Tn" COS p COS "V Li i 



rt \ 



/ g '(cos)= 



_ 



O L I L 



i i-/ . o /I\-T 



. 4 [(L - K-K~)~ 



' J 



The upper sign being taken for the xoiml (x = 0), the lower for the sectorial 

 harmonic (x = 2). 



If these expressions be developed in powers of K as far as three terms of the 

 series, I find, on re-introducing the factor f'~y~, 



(17), 



t 



. . (18). 



In " Harmonics" [ made the followin definitions 



In order to make the two forms of definition agree we must take 

 Thus 



. . (19). 



Now on development 



* 8 = (DM 1 - 5** + W*) = 



- 10/3 + i 



