Page:The Solar System - Six Lectures - Lowell.djvu/51



and equate it to that of the circle supposed described about that focus with the length of radius $$\scriptstyle \sqrt{ab} \,\!$$. This geometrically is the point of intersection of the two curves, since the value of r is common to both.

Consequently for the point sought whence, since

whence, since $$\scriptstyle b = a \sqrt{1-e^{2}} \,\!$$,

and $$\scriptstyle \cos v = \frac{(1-e^{2})^{\frac{3}{4}} - 1}{e} \,\!$$.

In the case of Mercury, e = .205605; v, the true anomaly of the point of maximum libration, is therefore 98° 55'.13.

But $$\scriptstyle \frac{a-r}{a \; e} = \cos E \,\!$$,

where E is the eccentric anomaly; and $$\scriptstyle E - e \sin E = M \,\!$$, where M is the mean anomaly; whence $$\scriptstyle v - M = \zeta \,\!$$, which is the amount of the maximum libration, is 23° 40' 38".

The gain or loss of the rotation over the revolution is the same thing as the equation of the centre.

We have, then, in the libration, a most conclusive and interesting proof of the isochronism of rotation and revolution.

The next point to consider is what caused this