THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
709 
From the last of these, 
I-J = 
1 
m + 1 I 
?i 
— k. 2 is the precession of the system of proper planes, and the above results show that 
the solar precession of the planet and satellite together, considered as one system, is 
one (m+i) th of the angular velocity which the nodes of the satellite would have, if the 
planet were spherical. 
k . 2 — k 1 is the lunar precession of the earth which goes on within the system, and it 
is approximately the same as though the sun did not exist. (Compare the second and 
fourth of (107) with N=xjj, and use (108)). 
It also appears that the lunar proper plane is inclined to the planet’s proper plane 
at a small angle the ratio of which to the inclination of the earth’s proper plane to the 
ecliptic is equal to one (ltl+l) th part of lt/J. 
If IT and l are of approximately equal speeds the proper plane of the moon will 
neither be very near the ecliptic, nor very near the earth’s proper plane. The results 
do not then appear to be reducible to very simple forms; nor are the angular velocities 
k 2 and k 2 — k x so easily intelligible, each of them being a sort of compound precession. 
If the solar influence were to wane, M and Q, the poles of the proper planes, would 
approach one another, and ultimately become identical, The two planes would have 
then become the invariable plane of the system ; and the two circles would be 
concentric and their radii would be inversely proportional to the two moments of 
momentum (whose ratio is in). 
Now in the problem which is to be here considered the solar influence will in effect 
wane, because the effect of tidal friction is, in retrospect, to bring the moon nearer 
and nearer to the earth, and to increase the ellipticity of the earth’s figure ; hence the 
relative importance of the solar influence diminishes. 
We now see that the problem to be solved is to trace these proper planes, from 
their present condition when one is nearly identical with the ecliptic and the other is 
the mean equator, backwards until they are both sensibly coincident with the equator. 
We also see that the present angular velocity of the moon’s nodes on the ecliptic is 
analogous to and continuous with the purely lunar precession on the invariable plane 
of the moon-earth system; and that the present luni-solar precession is analogous to 
and continuous with a slow precessional motion of the same invariable plane. 
Analytically the problem is to trace the secular changes in the constants of integra¬ 
tion, when a, a, /3, b, instead of being constant, are slowly variable under the influence 
of tidal friction, and when certain other small terms, also due to tides, are added to 
the differential equations of motion. 
