Page:The Solar System - Six Lectures - Lowell.djvu/94



outer edge of the system, supposing the density of the satellite to be $$\textstyle\frac{6}{5}$$ of the primary's. Now the satellites of Saturn are certainly a little denser than the planet. From our present values of its mass and volume, Titan's density comes out .24. This, then, is what has limited the system externally.

For the rest of it, another force has proved fashioner.

Our mathematics do not permit us to solve rigorously the problem of three bodies; that is, the motion of a first revolving round a second and perturbed by a third. We have to have recourse to approximations in series. We can thus determine to any degree of accuracy the result. Now the perturbative effect produced by a third body upon the major axis of a second revolving in its own plane may be expressed by a series developed in terms depending on powers of the eccentricity and cosines of multiple arcs of the mean motions. The typical form of one of these terms is

$$\textstyle P\frac{ca^{2}n}{pn-qn'}e^{l}e^{m}\cos{\overline{pn-qn'-Q}} $$

where $$P$$ is a function of $$a$$ and $$a'$$, the radii vectores.

From this, it appears that if $$p$$ and $$q$$ are nearly in the inverse ratio of the mean motions,

$$pn-qn'$$ is nearly $$o$$,