THE ELEMENTS OF THE ORBIT OP A SATELLITE. 
717 
A planet is attended by one or more satellites which raise frictional tides (either 
bodily or oceanic) in their planet; it is required to find the secular changes in the 
orbits of the satellites due to tidal reaction. 
This problem is however intimately l'elated to a consideration of the parallel changes 
in the inclination of the planet’s axis to a fixed plane, and in its diurnal rotation. 
It will therefore be necessary to traverse again, to some extent, the ground covered 
by my previous paper “ On the Precession of a Viscous Spheroid.” 
In the following investigation the tides are supposed to be a bodily deformation of 
the planet, but a slight modification of the analytical results would make the whole 
applicable to the case of oceanic tides on a rigid nucleus.* The'analysis will be such 
that the results may be applied to any theory of tides, but particular application will 
be made to the case where the planet is a homogeneous viscous spheroid, and the 
present paper is thus a continuation of my previous ones on the tides and rotation of 
such a spheroid. 
The general problem above stated may be conveniently divided into two :—- 
First, to find the secular changes in mean distance and inclination of the orbit of a 
satellite moving in a circular orbit about its planet. 
Second, to find the secular change in mean distance, and eccentricity of the orbit of 
a satellite moving in an elliptic orbit, but always remaining in a fixed plane. 
As stated in the introductory remarks, it will also be necessary to investigate the 
secular changes in the diurnal rotation and in the obliquity of the planet’s equator to 
the plane of reference. 
The tidally distorted planet will be spoken of as the earth, and the satellites as the 
moon and sun. 
This not only affords a useful vocabulary, but permits an easy transition from ques¬ 
tions of abstract dynamics to speculations concerning the remote history of the earth 
and moon. 
§ 2. Notation—Equation of variation of elements. 
The present section, and the two which follow it, are of general applicability to the 
whole investigation. 
© 
For reasons which will appear later it will be necessary to conceive the earth to 
have two satellites, which may conveniently be called Diana and the moon. The 
following are the definitions of the symbols employed. 
The time is t, and the suffix 0 to any symbol indicates the value of the correspond¬ 
ing quantity initially, when t — 0. The attraction of unit masses at unit distance is p. 
For the earth, let— 
M= mass in ordinary units; a= mean radius; w— density, or mass per unit 
volume, the earth being treated as homogeneous; g— mean gravity; fp/a; 
* Or, as to Part III., on a nucleus which is sufficiently plastic to adjust itself to a form of equilibrium. 
MDCCCLXXX. 4 Z 
