THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
8G7 
that found and discussed in the paper on “Precession,” where the plane of the lunar 
orbit was supposed to be coincident with the ecliptic. If the viscosity be small the 
equation reduces to a very simple form; this is given in (70). In § 10 I pass to 
case (ii.), where the earth and moon are the only bodies. The equation expressive of 
the rate of change of inclination of the lunar orbit to the invariable plane is given 
in (71). Fig. 5 illustrates the physical meaning of the equation, and an explanation of 
it is given in § 10. From it we learn that the effect of the tides is always to cause a 
diminution of the inclination—at least so long as the periodic time of the satellite, 
as measured in rotations of the planet, is pretty long. The following considerations 
show that this must generally be the case. It appears from the paper on “ Pre¬ 
cession ” that the effect of tidal friction is to cause a continual transference of moment 
of momentum from that of terrestrial rotation to that of orbital motion ; hence it 
follows that the normal to the lunar orbit must continually approach the normal to 
the invariable plane. It is true that the rate of this approach will be to some 
extent counteracted by a parallel increase in the inclination of the earths axis to the 
same normal. It will appear later that if the moon were to revolve very rapidly 
round the earth, and if the viscosity of the earth were great, then this counteracting 
influence might be sufficiently great to cause the inclination to increase.* This 
possible increase of inclination is not exhibited in fig. 5, because it illustrates the case 
where the sidereal month is 15 days long. 
In § 11 it is shown that, for case (ii.), the rate of variation of the mean distance, 
obliquity, and terrestrial rotation follow the laws investigated in “ Precession,” but 
that the angle, there called the obliquity of the ecliptic, must be interpreted as the 
angle between the plane of the lunar orbit and the equator. 
In § 12 I return again to case (i.) and find the laws governing the rate of increase 
of the obliquity of the ecliptic, and of decrease of the diurnal rotation of the earth. 
The results differ so little from those discussed in “ Precession ” that they need not be 
further referred to here. 
Up to this point no approximation has been admitted with regard to smallness 
either in the obliquity or the inclination of the orbit, but mathematical difficulties 
have rendered it expedient to assume their smallness in the following part of the 
paper. 
§ 31. Summary of Part III. 
Part III. is devoted to case (iii.) of our first problem. It was found necessary in 
the first instance to consider the theory of the secular inequalities in the motion of a 
moon revolving about an oblate rigid earth, and perturbed by a second satellite, the 
sun. The sun being large and distant, the ecliptic is deemed sensibly unaffected, and 
is taken as the fixed plane of reference. 
The proper plane of the lunar orbit has been already referred to, but I was here led 
* See the abstract of this paper, Proc. R.S., No. 200, 1879, for certain general considei’ations bearing 
on this case. 
