XVI.] 



MOTION IN AN ORBIT OF ANY INCLINATION. 



75 



Hence 



- q - 2mN* - m/ 3 -~ = - - m? cos i ; 



therefore as a first approximation 



hence 



q = - TO 2 cos 



* 4 



3 i 



- 2 ( 1 TO) /, = - m 2 sin * cos 2 - , 



3 m" 



2/2 = - TO 2 sin i cos i, 



2 ( m. + - m~ cos i j /, = - TO 2 sin i, 



and 



3 ... 



- m~ sin ^ cos 



T /-, \ T 3 , . . , i T 



2 ( 1 + TO) 1 4 = - 7?i sin ^ sin" - , 7 4 = 



1+7 TO COS i 



4 



3 TO 2 . . . i 

 - sin i sin 2 - , 



8 1 + TO 2 



3 ;- 



o T " " 

 8 1 m 



= - m' cos ^, 



/ 3 \ 3 



2 ( m + - TO 2 cos i ) ^V 3 - TO 2 cos i, 



v = - m" cos i, 



3 TO COS 1 



1 + 7 TO COS { 



4 



3 TO 2 . ,i 



Substitute above for the quantities I 3 , N 3 and we get the second approxi- 

 mation to q, 



3 .9 ., . 9 , . . . 

 q -m cos i TO cos' i + TO sin- 1. 



4 32 64 



It will be observed that I 3 , N 3 are of lower order than the other co- 

 efficients, so that in order to obtain them correctly to the same order as 



dff 



the others we were obliged to retain small terms in -5- arising from the 



variability of N. 



102 



