201 



a 2 - 2a/3 = J(l 4- cos u 8 - 2 sin\) - J sin < x (1 + 3 cos i x ) 



2ft- ft 1 



Multiplying this by the value of sin i, we have 



sin 2n - ft 1 . 



(a* - 2a/3) sin * = I (1 + cosi, 8 - 2 sin't,) sin 

 + 



\ (1 f cos i* - 2 sin 2 t 1 ) cos «i - £ sin 2 t l (1 + 3 cos «y j 



A, + £ 2 



— sin 2w - w,. 

 2ft - ft 1 



Also, considering the variations introduced into <p, we have 



cos*! A + ^ 2 . 



O = nt + -k— — — cos 2ft - ft 1 



x smtj 2ft -ft 1 



also 0 is the true longitude measured from the moveable axis whose 

 longitude measured backwards is ; therefore if n x t be the longitude 

 measured from a fixed axis, that measured from the moveable axis will 

 be nH + ; or, from the value of given above, 



ix 1 1 A, + B 2 , 



nH + - r cos 2ft - ft 1 , 



2 sin * x 2ft - w 1 



„ n , ,, , 2 COS*! - 1 + Z? 2 , 



2<p - 0 = 2nt - n l t + -k : -cos 2ft -ft 1 , 



r * smtj 2ft -ft 1 



and 



• rs 7 • n , 1 2 COS «j - 1 4, + i* 2 



sm 20 - 0 = sin 2ft - ft 1 + i r- 5 5 i. 



r * sm^ 2ft -ft 1 



Multiplying this by the value of (a 2 - 2 a/3) sin i, we shall have for the 

 constant term depending upon A x + _# 2 , 



j-Lfl+cos ^" 2 -2sin\)(3 cos i, - 1) - sin 2 *, (1 + 3 cos O } ^' + ^ 2 

 (lo V © ) 2ft - ft 1 



which may be put into the simpler form 



1 f 1 + cos i, -- sin\ (1+3 cos Al + B<1 



4\ 4 . v 'M 2ft -ft 1 



and in like manner the quantity (a 2 + 2a/3) sin « will contain the con- 

 stant quantity 



\ ( 1 + cos t t -4 sin 2 ^ (1+3 cos i,)] ^ — ^ 

 4 \ 4 v ,y / 2ft - ft 1 



