LECT. XIII.] THE MOTION OP THE APSE, ETC. 59 



Now we may put h = 1+ rj, where rj vanishes with e. Neglecting powers of 

 e above the first 



cl"n dh 



Neglect at first the last term : 



- 

 +m)esin(l -2m0 + ra-) + 3m 2 (l m) e sin (3 



9m-r) sin (2 2m) 0. 



_i_ iffi 



__ _ 



rj= 3m 2 - ecos(l 2m# + OT)-3m 2 f -e cos (3 -2m0 -w). 



1 ^m O ZiTtl 



Substitute this in the last term, and we get 



-^ = 3m 2 (l + m)e sin(l 2m0 + ra-) + 3m 2 (l m)esin(3 2m CT) 



27 4 2 + 4m -4m 2 ( . 

 " ~2 rrl (1 -2m) (3 -2m) e 



Now consider the other equation 

 <*1'_ H * _M I , , , 2 ^ /0_0>\ 



rr 1 Q 



= -3 + - mVr + - mVr [cos ( 2 2m) 6 2me cos ( 1 2m 6 + ar) 



7^ I Zi Zj 



+ 2me cos (3 2m CT)]. 



Differentiate the assumed expression for - , and let h be chosen so that 



the first differential coefficient shall have the same form as if h, e, OT were 

 constant. 



Thus 



1 dr 1 , . . dQ 1 . d9 



-j ji = 7i- (2 - 2m) a, sin (2 - 2m) 7 + -,-r- e sin (0 - nr) -y- , 

 r 1 a^ ha dt h?a ' dt 



where 



- T T^ [1 + 2 cos (2 2m) 0] + , x cos (0 - cr) + e -^ sin (0 CT) = 0, 



or 



-jf. cos (6 TO-) + e T;. sin(0 ?3-)= 6m 2 (l +m)e sin (l - 



6m 2 (l m) e sin (3 2m CT) 



4 2 + 4m 4m 2 

 (1- 2m) (3 -2m)' 



82 



