206 



PIERCE. 



Let us now decompose the coefficient of the first integral of (31) as 

 follows: 



1 . cos^G 1 



+ 



= - + cos2 G - 



cos- 



1 1 + cos 2 6' 



4 4 



cos2G 



+ cos2 G 



+ cos2 G. 



Then the whole equation (31) may be written 



P = 



— cos^ < — {d — ro) - 

 c { h ) 



cos 2 6' T'^^ (1 - cos t) t?7 



sm^B , sin2 5sin2^ 



+ 



4 



+ cos2 G 



i p^(l_ 



— COS 7) d'y 



sin 2 (9 C 

 4 Jo 

 sin 2 6 r 

 2 Jo 



4^ 



AA 



sm7 



7 

 sin 7 



(^7 



d-y 



(34) 



The various integrals may now be obtained by expanding in series 

 and integrating term by term. This gives 



V 



2/2 



COS'' 



Y (ct - '•0) j- 



sin 25 /sin 2 A 



2 



2A 



- 1 



_ cos 26 j (W _ (Ml' j_ MZ _ 

 4 ^ 2!2 4!4 6!6 



1 + cos 26 j (2^2 ^ (2^4 (2^)6 

 "^ 9 ) ';>'2 4!4 "^ fi'fi 



+ 



2 



sin 26 ^ 

 4 { 

 sin 26 



I 2!2 



4A - 



4!4 



(4/1)^ 

 3! 3 



+ 



6!6 



{4AI 

 5!5 



9A (2^)^ I (2^)^ 



3!3 5!5 



(35) 



Let us now eliminate B from the first terms of this equation, by 

 substituting B = G — A, obtaining 



