By Professor Fell. 11 



Let these equations be multiplied ^jfyf ft respec- 

 tively and added together ; and let all the terms disappear from 

 the resulting equation, except that involving x^. We must have 

 then 



, afr = /r-l + /r + 1 



This gives us 



/, = A cos (rS + B ) 

 where 



g' = 2 cos 3 

 The condition that x^ disappears is 



(2 cos 9 - l)/i=/2 

 whence B = - i 9 ; and supposing /i == 1, we have 

 cos (r - i)5 

 •^^ ^ cos i S 

 The corfficient of x^ in the resulting equation is 



(2 cos S - 1)/. - /_, = - ^^i^n^.in\^ 



cos i 9 

 and the equation is 



2 sin w 5 sin i 9 q 



cos i 3 ' 



In a similar manner we obtain 



, 2 sin 7i 9 sin i 9 



iTi = 



COS i 9 



and since 



2 cos 9. ar^ = ^JTr-l + ^r+l 



a?r = -4 COS {r ^ + B) 

 The first equation gives B = — ^9, and therefore 

 Xj== A cos (r — i) 9 

 a?j = ^ cos ^ fl 



_ cos (?- — i) 5 



*r r~7 — ^l 



cos i 6 



If we put ^ = y >re have 



sin 7? 9 = sin 9 (2 cos 9 — 2 cos 2 y) (2 cos 9—2 cos 4 y) 

 (2 cos 9 — 2 cos 2 {n — 1) y) 



2 sin n 9 sin -i- 9 . „ , „ . 



^^-^ =4 sin^ 19 (- 4sin^ | 9 + 4 sin^ y) (-4 sin' \ Q 



2 9 



COS 



+ 4 sin^ 2 y) 



