205 



even where it is not, it will be best to reserve them, and, if necessarj', 

 to take them into account after lower powers have been examined. 



The other, and more important way in which such terms arise is as 

 follows: — Take the value already found for <p 1} viz. — 



1 cos i, (A x - B: 2 A,-B 2 



nt + : — < , cos 2n - n 1 + = — cos n 1 



2 1 sin i x \2n - w n l 



A 2 -B 1 . A 2 + B, . ) 



- - sm 2n - n 1 sin w) 



2n - n 1 n x ) 



this will give 



1 costj \f A y - B 2 A x + Z? 2 



cos 0 = cos n 



4 1 sin iy V »' 2w - n 



A 2 -B x A 2 +B x 



sm n - n L - \ + ; — | cos n - w 



\ 2n - n 1 n l 



Multiply this by = A x cos n - n 1 + B x sin n - n\ and retain the terms 

 involving cos 2n - 2n l , and we have 



1 cos*! If A, -B 2 A l +B 2 \ 



(V y COS 0 = : { - B x 



v 8 1 sin i x \\ n 1 2n-n l j 



, A 2 - B x A. 2 + B x . 



+ ( r + , \A X ] cos 2n - 2n l 



2n-n n L 



in like manner we shall find 



. , lcosti (f A X + B* A x -B 2 . 

 w 2 sm 0= - i -i 2- + - A 2 



fi cin i S\ V/M vi 1 vi i / 



8 sin t, \\ 2n - n x n 1 



A 2 -B x _ A 2 + B 

 2n - n 1 n x 



B 2 \ cos 2n - 2n l 



di , . j lcosi \A X -B 2 A 2 -B X A x + BoA 2 + B x 



•'• T^ = w > cos 0 - w 2 sm 0 = - - — J + — 



at 8 sin i I 2n - n l n l 



_ A 2 -B x A x -B 2 _ A 2 + B X A X + B 2 \ CQg 2 ^_ 2 ^ 

 n 1 2n - n 1 ) 



1 cost ,— — 2n-2n x 



= n-r- (A 2 + B x A X + B 2 -A X -B 2 A 2 -B\) — — cos2rc-2% 



o sm * (2n - n 1 ) n 1 



1 cos 



- — \A 2 + B X A x + B 2 - A x -B 2 A 2 -B x 



- — - sin (2n - 2m 1 ) 



{2n - n x ) n 



E. I. A. PROC. VOL. X. 2F 



