ATTENDED BY SEVERAL SATELLITES. 
495 
planet’s rotation and to the orbital motion of all the satellites. And let CE be the 
whole energy, both kinetic and potential, of the system. Then h is the angular 
velocity with which the planet would have to rotate in order that the rotational 
momentum might be equal to that of the whole system ; and E is twice the square of 
the angular velocity with which the planet would have to rotate in order that the 
kinetic energy of planetary rotation might be equal to the whole energy of the system. 
By the principle of conservation of momentum h is constant, and since the system is 
non-conservative of energy E is variable, and must diminish with the time. 
The kinetic energy of the orbital motion of the satellite m is ^jxMrn/c, and the 
potential energy of position of the planet and satellite is —/xJTm/c ; the kinetic 
energy of the planet’s rotation is \Cn 2 . Thus we have, 
Ch=Cn +S 
(M+mf' 
( 3 ) 
2 CE=Crv l -t 1 ^ .(4) 
In the equations (3) and (4) we may regard C as a constant, provided we neglect 
the change of ellipticity of the planet’s figure as its rotation slackens. 
Let the symbol b indicate partial differentiation; then from (3) and (4) 
bn 1 fj}Mm 
bid 1 ) G (41 + to) 4 
bE _ 1 fj/Mm 1 fjbM'ni 
~b(dj = G (M+mf ll ~C c 1 
But 
1 /iMm 1 /JMm 
C W 1 ” = C (M+my 
i j-i e C(M+m) h bE 
and therefore - —rzrz — =n—/2.( 0 ) 
fEMm t(d) v ’ 
From equations (2) and (5) we may express the rate of increase of the square root 
of any satellite’s distance in terms of the energy of the whole system, in the general 
case where the planet has any degree of viscosity. A good many transformations, 
analogous to those below, may be made in this general case, but as I shall only 
examine in detail the special case in which the viscosity is small, it will be convenient 
to make the transition thereto at once. 
When the viscosity is small, p, which varies inversely as the viscosity, is large. 
Then, unless n — f2 be very large, (n— 12)/jp is small compared with unity. Thus in (2) 
we may neglect (n — fl ) 2 /p 3 in the denominator compared with unity. 
Substituting from (5) in (2), and making this approximation, we have 
fj)Mm dc h 
(df+m) 1 dt 
. 3 .2 C (m/i) 2 bE 
1 Q c 6 b(c-) 
(6) 
