THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
715 
It will probably conduce to the intelligibility of what follows, if an explanatory out¬ 
line of the contents of the paper is placed before the reader. Such an outline must of 
course contain references to future procedure, and cannot therefore be made entirely 
intelligible, yet it appears to me that some sort of preliminary notions of the nature 
of the subject will be advantageous, because it is sometimes difficult for a reader to 
retain the thread of the argument amidst the mass of details of a long investigation, 
which is leading him in some unknown direction. 
Part VIII. contains a general review of the subject in its application to the evolu¬ 
tion of the planets of the solar system. This is probably the only part of the paper 
which will have any interest to the general reader. 
The mathematical reader, who merely wishes to obtain a general idea of the results, 
is recommended to glance through the present introduction, and then to turn to 
Part VII., which contains a summary, with references to such parts of the paper as it 
was not desirable to reproduce. This summary does not contain any analysis, and 
deals more especially with the physical aspects of the problem, and with the question 
of the applicability of the investigation to the history of the earth and moon, but of 
course it must not be understood to contain references to every point which seems to 
be worthy of notice. I think also that a study of Part VII. will facilitate the compre¬ 
hension of the analytical parts of the paper. 
Part I. contains an explanation of the peculiarities of the method of the disturbing 
function as applied to the tidal problem. At the beginning there is a summary of the 
meaning to be attached to the principal symbols employed. The problem is divided into 
several heads, and the disturbing function is partially developed in such a way that it 
may be applicable either to finding the perturbations of the satellite, or of the jffanet 
itself. 
In Part II. the satellite is supposed to move in a circular orbit, inclined to the fixed 
plane of reference. It here appears that the problem may be advantageously sub¬ 
divided into the following cases : 1st, where the permanent oblateness of the planet is 
small, and where the satellite is directly perturbed by the action of a second large and 
distant satellite such as the sun ; 2nd, where the planet and satellite are the only two 
bodies in existence ; 3rd, where the permanent oblateness is considerable, and the 
action of the second satellite is not so important as in the first case. The first and 
second of these cases afford the subject for the rest of this part, and the laws are found 
which govern the secular changes in the inclination and mean distance of the satellite, 
and the obliquity and diurnal rotation of the planet. 
Part III. is devoted to the third of the above cases. It was found necessary first 
to investigate the motion of a satellite revolving about a rigid oblate spheroidal planet, 
and perturbed by a second satellite. Here I had to introduce the conception of a pair 
of planes, to which the motions of the satellite and planet may be referred. The 
problem of the third case is then shown to resolve itself into a tracing of the secular 
changes in the positions of these two “ proper ” planes, under the influence of tidal 
