THE ELEMENTS OF THE ORBIT OF A SATELLITE. 
871 
degree, at least wlien the day is not very nearly equal to the sidereal month. When 
that relation between the day and month is very nearly fulfilled, the equation may 
become largely affected by the viscosity; and I shall return to this point later, while 
for the present I shall assume the equation to give satisfactory results. 
This equation of conservation of moment of momentum enables us to compute as 
many parallel values of the day ancl month as may be desired. 
Now we have got the time-rates of change of the inclinations of the lunar orbit to 
its proper plane, and of the earth’s proper plane to the ecliptic, and we have also the 
time-rate of change of the square root of the moon’s distance. Hence we may obtain 
the square-root-of-moon’s-distance-rate (or shortly the distance-rate) of change of the 
two inclinations. 
The element of time is thus entirely eliminated; and as the period of time required 
for the changes has been adequately considered in the paper on “ Precession,” no 
further reference will here be made to time. 
In a precisely similar manner the equations giving the time-rate in the cases (i.) and 
(ii.) of our first problem, may be replaced by equations of distance-rate. 
Up to this point terrestrial phraseology has been used, but there is nothing which 
confines the applicability of the results to our own planet and satellite. 
§ 32. Summary of Part IV. 
We now, however, pass to Part IV., which contains a retrospective integration of 
the differential equations, with special reference to the earth, moon, and sun. The 
mathematical difficulties were so great that a numerical solution was the only one 
found practicable.* The computations made for the paper on “Precession” were used 
as far as possible. 
The general plan followed was closely similar to that of the previous paper, and 
consists in arbitrarily choosing a number of values for the distance of the moon from 
the earth (or what amounts to the same thing for the sidereal month), and then 
computing all the other elements of the system by the method of quadratures. 
The first case considered is where the earth has a small viscosity. And here it may 
be remarked that although the solution is only rigorous for infinitely small viscosity, 
yet it gives results which are very nearly true over a considerable range of viscosity. 
This may be seen to be true by a comparison of the results of the integrations in 
§§15 and 17 of “ Precession,” in the first of which the viscosity was not at all small; 
also by observing that the curves in fig. 2 of “ Precession ” do not differ materially from 
the curve of sines until e (the f of this paper) is greater than 25° ; also by noting a 
similar peculiarity in figs. 4 and 5 of this paper. The hypothesis of large viscosity 
does not cover nearly so wide a field. 
* An analytical solution in the case of a single satellite, where the viscosity of the planet is small, is 
given in Proc. Roy. Soc., No. 202, 1880. 
5 T 2 
