In the [following figure], P is the pole of the earth; E the pole of the ecliptic; L the pole of the lunar orbit; V the mean position of the pole of the vortex at the time; the angle ♈EL the true longitude of the pole of the lunar orbit, equal to the true longitude of the ascending node ± 90°. VL is therefore the mean inclination ± 2° 45′; and the little circle, the orbit described by the pole of the vortex twice in each sidereal revolution of the moon. The distance of the pole of the vortex from the mean position V, may be approximately estimated, by multiplying the maximum value 15′ by the sine of twice the moon’s distance from the node of the vortex, or from its mean position, viz.: the true longitude of the ascending node of the moon on the ecliptic. From this we may calculate the true place of the node, the true obliquity, and the true inclination to the lunar orbit. Having indicated the necessity for this correction, and its numerical coefficient, we shall no longer embarrass the computation by such minutiæ, but consider the mean inclination as the true inclination, and the mean place of the node as the true place of the node, and coincident with the ascending node of the moon’s orbit on the ecliptic.

POSITION OF THE AXIS OF THE VORTEX.

It is now necessary to prove that the axis of the vortex will still pass through the centre of gravity of the earth and moon.

Let XX now represent the axis of the lunar orbit, and C the centre of gravity of the earth and moon, X′X′ the axis of the vortex, and KCR the inclination of this axis. Then from

similarityCt:Tt: :Cm:Mm
butTt:Mm: :Moon’s mass:Earth’s mass.
That isTt:Mm: :TC:MC.

Therefore the system is still balanced; and in no other point but the point C, can the intersection of the axes be made without destroying this balance.

It will be observed by inspecting the [figure], that the arc R′K′ is greater than the arc RK. That the first increases the arc AR, and the second diminishes that arc. The arc R′K′ is a plus correction therefore, and the smaller arc RK a minus correction. If the moon is between her descending and ascending node, (taking now the node on the ecliptic,) the correction is negative, and we take the smaller arc. If the moon is between her ascending and descending node, the correction is positive, and we take the larger arc. If the moon is 90° from the node, the correction is a maximum. If the moon is at the node, the correction is null. In all other positions it is as the sine of the moon’s distance from the nodes. We must now find the maximum value of these arcs of correction corresponding to the mean inclination of 2° 45′.

To do this we may reduce TC to Tt in the ratio of radius to cosine of the inclination, and taking TS for radius.