THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
869 
the precessional velocity, the proper plane of the satellite is inclined at a small angle 
to the equator of the planet. The rapidity of motion of the satellites of Mars, 
Jupiter, and of some of the satellites of Saturn, and their smallness compared with 
their planets, necessitates that their proper planes should be inclined at small angles 
to the equators of the planets. A system may, however, be conceived in which the 
two proper planes are inclined at a small angle to one another, but where the 
satellite’s proper plane is not inclined at a small angle to the planet’s equator. 
In the case now before us the regression of the common node of the two proper 
planes is a sort of compound solar precession of the planet with its attendant moon, 
and the regression of the two nodes on their respective proper planes is very nearly 
the same as the purely lunar precession on the invariable plane of the system. Thus 
there are two precessions, the first of the system as a whole, and the second going on 
within the system, almost as though the external precession did not exist. 
If the nodal velocity be of nearly equal speed with the precessional velocity, the 
regression of the proper planes and of the nodes on those planes are each a compound 
phenomenon, which it is rather hard to disentangle without the aid of analysis. Here 
none of the angles are necessarily small. 
It appears from the investigation in “ Precession ” that the effect of tidal friction is 
that, on tracing the changes of the system backwards in time, we find the moon getting 
nearer and nearer to the earth. The result of this is that the ratio of the nodal 
velocity to the precessional velocity continually diminishes retrospectively ; it is 
initially very large, it decreases, then becomes equal to unity, and finally is very 
small. Hence it follows that a retrospective solution will show us the lunar proper 
plane departing from its present close proximity to the ecliptic, and gradually passing 
over until it becomes inclined at a small angle to the earth’s proper plane. 
Therefore the problem, involved in the history of the obliquity of the ecliptic and in 
the inclination of the lunar orbit, is to trace the secular changes in the pair of proper 
planes, and in the inclinations of the orbit and equator to their respective proper planes. 
The four angles involved in this system are however so inter-related, that it is only 
necessary to consider the inclination of one proper plane to the ecliptic, and of one 
plane of motion to its proper plane, and afterwards the other two may be deduced. I 
chose as the two, whose motions were to be traced, the inclination of the lunar orbit to 
its proper plane, and the inclination of the earth’s proper plane to the ecliptic ; and 
afterwards deduced the inclination of the moon’s proper plane to the ecliptic, and the 
inclination of the equator to the earth’s proper plane. 
The next subject to be considered (§ 14 to end of Part III.) was the rate of change 
of these two inclinations, when both moon and sun raise frictional tides in the earth. 
The change takes place from two sets of causes :— 
First because of the secular changes in the moon’s distance and periodic time, and in 
the earth’s rotation and ellipticity of figure—for the earth must always remain a figure 
of equilibrium. 
MDCCCLXXX. 5 T 
