Page:The Solar System - Six Lectures - Lowell.djvu/110



plane of the rotating planet, one preceding the position of the tide-raising body, the other diametrically opposite.

The action of these ansæ upon the attracting body would be analogous to, but in one vital respect different from, that of an equatorial protuberance. Like that, they would tend to alter the position of the axis of rotation at right angles to the pull upon them, but the pull being always backward the axis is constantly solicited forwards toward the attracting body. Consequently the axis of rotation, while rotating round the axis of the orbit, would generally seek the satellite. For the force here, when the axes are perpendicular, is at its maximum. The axis, therefore, continues to tend toward the orbital plane.

Analytically, in this case, unlike that of an equatorial bulge due to axial rotation, the expression for the change of inclination contains a term dependent on the time and increasing with it.

This term causes the inclination of the equatorial to the orbital plane to diminish until the axis of rotation lies in the plane of the orbit.

The tidal force varies as $$\textstyle \frac{2mr}{d^{3}}$$, approx., and its work for any given time as $$\textstyle \frac{4m^{2}r^{2}}{d^{6}}$$, approx.

It should therefore be much more potent upon