1879.] On Secular Changes in the Orbit of a Satellite. 5 



planet depends npon the position of the satellite, therefore the 

 elements of the satellite's orbit will appear in the disturbing function, 

 :as representing the state of tidal disturbance in the planet ; but these 

 elements also appear as representing the position of the satellite as a 

 body whose motion is perturbed. In order to apply the method of 

 the disturbing function they ought only to enter in the latter capacity. 

 This difficulty is overcome by supposing the earth to have two 

 satellites; one the tide-raising satellite (called Diana in the memoir), 

 and the other the perturbed satellite or moon. Then, after the appli- 

 cation of the method of the disturbing function, the tide-raising 

 satellite is made identical with the moon; and thus the effect of the 

 lunar tides on the moon's motion is determined. Or else the tide- 

 raising satellite is made identical with the sun, so as to find the 

 effect of the solar tides on the moon. The method of the disturbing 

 function is also applied to determine the perturbations of the earth's 

 rotation, and a similar artifice has to be used, because the earth has 

 to be treated in two capacities, first as a body in which tides are 

 raised, and secondly, as a body whose rotation is perturbed. 



The problem is divided into the two following cases : — 



1st. Where the lunar orbit is circular, but inclined to the ecliptic. 



2nd. Where the orbit is elliptic, but coincident with the ecliptic. 



The previous paper on " Precession," dealt with the mean distance 

 of the moon, and with the rotation of the earth and the obliquity of 

 the ecliptic ; therefore, in the present paper the inclination and eccen- 

 tricity afford the principal topics. 



The first of these problems occupies the larger part of the paper. 

 If the satellite and planet be the only bodies in existence, the problem 

 <of the inclination is not very complicated. 



The following considerations (in substitution for the analytical 

 treatment of the paper) will throw some light on the general effects of 

 tidal friction : — 



Suppose the motions of the planet and of its solitary satellite to be 

 referred to the invariable plane of the system. The axis of resultant 

 moment of momentum is normal to this plane, and the component 

 rotations are that of the planet's rotation about its axis of figure, and that 

 of the orbital motion of the planet and satellite round their common 

 centre of inertia ; the axis of this latter rotation is clearly the normal 

 to the satellite's orbit. Hence the normal to the orbit, the axis of 

 resultant m. of m., and the planet's axis of rotation, must always lie in 

 one plane. From this it follows that the orbit and the planet's equator 

 must necessarily have a common node on the invaluable plane. 



If either of the component rotations alters in amount or direction, a 

 corresponding change must take place in the other, such as will keep 

 the resultant m. of m. constant in direction and magnitude. 



It appears from the previous papers that the effect of tidal friction 



