THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
865 
paper. For convenience of diction I shall henceforth speak of the planet as the earth, 
and of the satellites as the moon and sun ; for, as far as regards tides, the sun may be 
treated as a satellite of the earth. The investigation has been kept as far as possible 
general, so as to be applicable to any system of tides in the earth ; but it has been 
directed more especially towards the conception of a bodily distortion of the earth’s 
mass, and all the actual applications are made on the hypothesis that the earth is a 
viscous body. A very slight modification would however make the results applicable 
to frictional oceanic tides on a rigid nucleus (see § 1 immediately after (15)). 
I thought it sufficient to consider the problem as divisible into the two following 
cases :— 
1st. Where the moon’s orbit is circular, but inclined to the ecliptic. (Parts I., II., 
III., IV.) 
2nd. Where the orbit is eccentric, but always coincident with the ecliptic. (Parts I., 
V., VI.) 
Now that these problems are solved, it would not be difficult, although laborious, to 
unite the two investigations into a single one ; but the additional interest of the 
results would hardly repay one for the great labour, and besides this division of the 
problem makes the formulas considerably shorter, and this conduces to intelligibility. 
For the present I only refer to the first of the above problems. 
It appears that the problem requires still further subdivision, for the following 
reasons :— 
It is a well-known result of the theory of perturbed elliptic motion, that the orbit of 
a satellite, revolving about an oblate planet and perturbed by a second satellite, always 
maintains a constant inclination to a certain plane, which is said to be r pro r per to the 
orbit; the nodes also of the orbit revolve with a uniform motion on that plane, apart 
from “periodic” inequalities. 
If then the moon’s proper plane be inclined at a very small angle to the ecliptic, the 
nodes revolve very nearly uniformly on the ecliptic, and the orbit is inclined at very 
nearly a constant angle thereto. In this case the equinoctial line revolves also nearly 
uniformly, and the equator is inclined at nearly a constant angle to the ecliptic. 
Here then any inequalities in the motion of the earth and moon, which depend on 
the longitudes of the nodes or of the equinoctial line, are harmonically periodic in 
time (although they are “secular inequalities”), and cannot lead to any cumulative 
effects which will alter the elements of the earth or moon. 
Again, suppose that the moon and earth are the only bodies in existence. Here 
the axis of resultant moment of momentum of the system, or the normal to the inva¬ 
riable plane, remains fixed in space. The component moments of momentum are those 
of the earth’s rotation, and of the moon’s and earth’s orbital revolution round their 
common centre of inertia. Hence the earth’s axis and the normal to the lunar orbit 
must always be coplanar with the normal to the invariable plane, and therefore the 
orbit and equator must have a common node on the invariable plane. This node 
