( 376 ) 



If wc miiltij»lv (he 2"'' member of this equation by ros 6 -\- cus tp 

 we find 



^ i X<^os6 = ><m{2H-l)(p^ ::£ {■l)i'[dn{;2n-p-\-l)(f-]^i<in{2n-p-\)tf^cosp6-\- 

 '^ I -\- siinfcos:2nO « 



^ j 2h— I 



1 f y^cos(f=-bin2ncfcos(f^2£{-\)l>—'^\sin[2ii-i)^\)if^sin(2.n-p-\)(f\cosp6\ 

 ^ p—i 



SO the sum becomes 



sin {2/1 — l)(f — sin 2n (f cos y -| sin (f cos 2n 8 = si)i tp {cos 2n 6 — cos 2n <f). 



From the formula {a) follows immediately 



sin 2n (p 



Aou = — 2jt —, . 



sin (p 



If we replace this value in the expressions just found, we arrive at 



i^ =: ; [/j sin 2<p -\- I^ sin itp -\- I^ sin G<^ -j- • • 1- 



sin ip 



Q^----^ [(/, - I,) sin 2<p + (/, - /,) sin 4r/ + . . ] 

 sin (p 



= 4rr [/j cos (p -\- /, cos 3(p -\- I^ cos htp . . ], 



R z= — [/, sin 2 ip — [^ sin 4<p -{- I^ sin 6(p — . . ], 



si7i <p 



2jt 



S =: — —, [/j sin 2(p — /., {sin 2<p -\- sin i(p) -\~ I. {sut 4(p -\- sm 69 ) . .], 



sin <p 



4.T cos <ƒ) . ,■-,,,•- 



— ^ [ Jj sDi (p — 7^ sin o(p -\- J . sin o(p — . . J. 



sin <p 



Moreover from the formnla (a) we can deduce anothei result : 

 When we develop 



cos 2nd — cos 2n(p 



— z=: 1 a„ -f a^ cos 6 -f ((, cos 26 A^ . . . 



cos tf -\- COS ip 



then we know that 



2 r cos 2n6 — COS 2nip 



a^ = — I — cos pd dd 



jr J cos 6 -\- cos ip 

 u 



If we compai'c this to the equation (r/) we arrive at 



,;_i sin {2n — p)ip 



a ={—l) 2 ; 



sin tp 



for jj = 0, J, 2, . , (2// — J), whilst for greater \alues of j) we have 

 a,j := 0. 



