THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
719 
Then let 
C 
l ~p,Mm n ° GQ .^ 
C 2v 
(For a homogeneous earth ——= —, and / 2 0 c 0 = 
(gody +1 I Ai 4 - Thus if we put 
©■“+->]' 
(6) 
k=sfi Q i ..(7) 
,s‘ is a time, being about 3 hrs ‘ 4 l mm - for the homogeneous earth. Jc is also a time, being 
about 57 minutes, with the present orbital angular velocity of the moon, and the earth 
being homogeneous). 
Then since f2=J2 0 £~ 3 , c=c 0 £' 2 , therefore 
C k 
flc =- 
liMm £ 
Again, since (c/c 0 )*=£ therefore 
1 ch_ 2 
c dt £ dt 
and since 77 = 1 — y/1 —e 3 , therefore 
dr/ e de 
dt \/l— e 2 dt 
Then substituting for R in terms of W in the four equations (1—4), and using the 
transformations ( 8 - 10 ), we get, 
d%_ dW 
.. — K, , 
dt de 
( 8 ) 
A) 
( 10 ) 
dr) _ k / dW dW\ 
clt~~ zylk+lryj 
and if the orbit be circular, so that e= 0 , dW/dzz=0, 
dj _ k / 1 dW 
dt % \siii j dN 
+ tan 
.dW 
de 
( 12 ) 
(13) 
. . dN k dW 
J dt f dj 
(14) 
These are the equations of variation of elements which will be used below. The 
last two (13) and (14) will only be required in the case where the orbit is circular. 
