716 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
friction. After a long analytical investigation differential equations are found for the 
rate of these changes. 
Part IV. contains the numerical integration of the differential equations of Parts II. 
and III, in application to the case of the earth, moon, and sun, the earth being 
supposed to be viscous. 
Part Y. contains the investigation of the secular changes of the eccentricity of 
the orbit of a satellite, together with the corresponding changes in the planet’s 
mode of motion. 
Part VI. contains a numerical integration of the equations of Part V. in the case of 
the earth and moon. The objects of Parts VII. and VIII. have been already explained. 
In the abstract of this paper in the Proceedings of the Royal Society,'"' certain 
general considerations are adduced which throw light on the nature of the results here 
found. This general reasoning is not reproduced here, because it is incapable of leading 
us to definite results, and it was only used there as a substitute for analysis. 
I. 
THE THEORY OF THE DISTURBING FUNCTION. 
§ 1. Preliminary considerations. 
In the theory of disturbed elliptic motion the six elements of the orbit may be 
divided into two groups of three. 
One set of three gives a description of the nature of the orbit which is being 
described at any epoch, and the second set is required to determine the position of the 
body at any instant of time. In a speculative inquiry like the present one, where we 
are only concerned with very small inequalities which would have no interest unless 
their effects could be cumulative from age to age, so that the orbit might become 
materially changed, it is obvious that the secular changes in the second set of elements 
need not be considered. 
The three elements whose variations are not here found are the longitudes of the 
perigee, the node, and the epoch; but the subsequent investigation will afford the 
materials for finding their variations if it be desirable to do so. 
The first set of elements 'whose secular changes are to be traced are, according to 
the ordinary system, the mean distance, the eccentricity, and the inclination of the 
orbit. We shall, however, substitute for the two former elements, viz.: mean distance 
and eccentricity, two other functions which define the orbit equally well; the first of 
these is a quantity proportional to the square root of the mean distance, and the 
second is the ellipticity of the orbit. The inclination will be retained as the third 
element. 
The principal problem to be solved is as follows:— 
* No. 200, 1879. 
