ATTENDED BY SEVERAL SATELLITES. 
497 
Let 
X= 
fjbM 1 rn 
' C(M+ mfa. . 
X is different for each satellite and is subject to suffixes 1, 2, 3, &c. 
On comparing (12) and (14) we see that 
X 
(14) 
—LLe 3 
(15) 
This is of course merely a form of writing the equation 
to)=/2 2 c 3 
A 
Then (4) may be written 
‘2,E=n*—%- 
(16) 
[In order to compute k and X we may pursue two different methods. 
First, suppose a — a, the planet’s mean radius. 
Then 
mot 
~C 
rWk; 
' ?va 2 r w ' 
M 1 
v 
i+v 
(M+7rCy 
/c=f[v 4 (l + ^) 3 ] _f (- ), of same dimensions as an angular velocity. 
\a 
i_i (9 
X = f[^ 6 (l + »>)] } J, of same dimensions as the square of an angular velocity. 
If v be large compared with unity, as is generally the case, the expressions become 
5 to / n 
K ~ 2 M V a* 
X=^ 
2M\a 
(17) 
Secondly, suppose M large compared with all the to’s, and suppose for example that 
the solar system as a whole is the subject of investigation. Then take a as the earth’s 
present radius vector, and oj as its present mean motion, and 
m ,—i x m u-Mm 
k— —y/ [xAla, and X= - 
0 ' 
or K—nn 
C 
C a 
X —TO 
C 
(18) 
C is here the sun’s moment of inertia.] 
Then collecting results from (9), (13), (16), the equations which determine the 
changes in the system are 
^—_ A — 1 
dt A 
and a similar equation for each satellite 
n~ h —Hkx 
X 
y. 
■ . . (19) 
2E=n 9, —% 
j 
