LECT. XVI.] MOTION IN AN ORBIT OF ANY INCLINATION. 73 



Again, we have the equation 



dt 

 which may be written 



dH^ _ 3 4 r . ,Q _ COS 41 + sin 2 2 sin2 1 Cos2 * + sin 2 /0 + ff\ sin 4 *~| 



7 O V / O / / X 



Substitute for /io- its approximate value n"a* in the small terms ; and we 

 find from these two equations 



3 i 3 i 



-(I -w)yl 1 + 4(l -m) 3 A 1 + -in"(l-m) cos 4 - = --m 2 cos 4 -, 



3 i i 3 i i 



- A, + 4 J 2 + ' r m 2 . 2 cos 2 - sin - = - - m" . 2 cos 2 - sin 2 - , 



2 222 22 



3 i i 



mA 3 + 4:m 3 A 3 + -m 3 . 2 cos 2 - sin 2 - = 0, 



3 , . t i 3 , . 4 i 



4 4 + -m 2 (l+m)sm 4 - = --m-sm 4 -. 



Therefore 



3 , 4 i 2-m 



l ~ ~ n a 



_ _ 

 2(l-m)(l-2m)(3-2m)' 



A,,= 2m 1 cos 2 - sin 2 - , 



Li Li 



A,= 3m 2 cos 2 -sin 2 - , - 



2 2(l-2m)(l+2m) J 



3 , . .i 2 + m 



,= -m sm 4 - 



and 



.H" 2 = n 2 a 4 1 1 + wi a - 1 m 2 (cos 4 | + sin 4 1 



I \ t 



^ _ C os 4 - cos 2(0 0') + 3m cos 2 - sin 2 - cos 20 + - - sin 4 = cos 2 



where a is defined by 



A. II. tL ~ n0 " 10 



