THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
887 
Here Cn is the moment of momentum of the planet’s rotation, and C£(l— y)/k is the 
moment of momentum of the orbital motion; and the whole moment of momentum is 
the sum of the two. 
£_ /jbMm 
Jc \/ /x(7LT T - in') 
between unit masses at unit distance. 
By a proper choice of units we may make fxMm /and C ecpial to unity."' 
Then let x be equal to the square root of the satellite’s mean distance c, and the 
equation of conservation of moment of momentum becomes 
n-\-x(l—r))=h .(a) 
If in (a) rj, the ellipticity of the orbit, be zero, we have equation (3) of the previous 
paper, No. 197, 1879. 
It is well known that the sum of the potential and kinetic energies in elliptic motion 
is independent of the eccentricity of the orbit, and depends only on the mean distance. 
Hence if CE be the whole energy of the system, we have (as in equations (2) and (4) 
of the above paper, No. 197), with the present units 
2E = n 3 —\ 
ar 
Then if z be written for 2E, and if the value of n be substituted from (a), -we have 
z —{h—x( 1—p)} —^.(£) 
This is the equation of energy of the system. 
-v/c, where /x is the attraction 
By the definitions of £ and h in § 2, C 
* In the paper above referred to, and in another, Proc. Roy. Soc., .No. 202, of 1880, the physical 
meaning of the units adopted is scarcely adequately explained. 
The units are such that C, the planet’s moment of inertia, is unity, that ) is unity, and that a 
quantity called s and defined in ( 6 ) of this paper is unity. 
From this it may be deduced that the unit length is such a distance that the moment of inertia of 
planet and satellite when at this distance apart about their common centre of inertia is equal to the 
moment of inertia of the planet about its own axis. If 7 be this unit of length, this condition gives 
M+^~ =C ’ or r <= 
C (M+m) 
Mm 
The unit of time is the time taken by the satellite to describe an arc of 57 0- 3 in a circular orbit at 
distance 7 ; it is therefore ( S. Y ^C The unit of mass is 
\jiMmJ \ Mm ) M+m 
From this it follows that the unit of moment of momentum is the moment of momentum of orbital 
motion when the satellite moves in a circular orbit at distance 7 . The critical moment of momentum of 
the system, referred to in those two papers and below in this appendix, is 4/3* of this unit of moment of 
momentum. 
5x2 
