XIII.] THE MOTION OF THE APSE, ETC. 61 



The terms in these two expressions which are independent of e give no 

 new information ; equating the others : 



- (2 - 2m) hf sin (2 - 2m) sin. (0 - ar) - i - (2 - 2m) a, sin (2 - 2m) 



+ [1 + h, cos (2 2m) 0J T^ sin (0 ra-) e ^ cos (9 w) 



3 3 F 



- m 2 e cos (6 TT) + ~ m 2 6a,e cos (0 cr) 



z ^j 



~1 

 e cos(3~2m0-nr) 



J 



3 



+ 9m 2 ?? cos (2 - 2m) 0. 

 Reducing this expression 



. - v /3 3 A //3 



, > sin OT + e ~ cos (t/ T&) = I - m - m 4 ) e cos (v m) 



, >j j~ Q - 



Cw (lu \Z 



CO 



3 + - : -m 4 )ecos(l 



+ ( 3 m 2 3m 3 + m 4 ) e cos (3 - 2mO 



\ 8 / 



and from before 



-,7; cos (9 zr) + e -Tn sin (^ CT) 1 8m 4 e sin (6 



dO 



+ (6m 2 + 6m 3 ) e sin ( 1 - 2m + m) 



+ (6m 2 - 6m 3 ) e sin (3 - 2m0 - nr). 

 Hence, 



3 , 147 



- m 2 H m 4 



4 16 



27 \ 



m 2 - 3m 3 - 3 m 4 j e sin (2 - 2m) 6 



(-1C Q /? O \ 



-m 2 -m 3 -- m 4 )esin(2m0 2sr) 

 4 2 16 / 



/ 3 3 15 4 



+ ( o m o m Tc r 



\ ^ ^ J- O 



