By whatever means the two planes first became permanently inclined, we
see that it is a necessary consequence of the admission of these
principles, not only that the axis of the vortex should be drawn aside
by the momentum of the earth and moon, ever striving, as it were, to
maintain a dynamical balance in the system, in accordance with the
simple laws of motion, and ever disturbed by the action of gravitation
exerted on the grosser matter of the system; but also, that this axis
should follow, the axis of the lunar orbit, at the same mean
inclination, during the complete revolution of the node. The mean
inclination of the two axes, determined by observation, is 2d 45', and
the monthly equation, at a maximum, is about 15', being a plus
correction in the northern hemisphere, where the moon is between her
descending and ascending node, reckoned on the plane of the vortex, and
a minus correction, when between her ascending and descending node. And
the mean longitude of the node will be the same as the true longitude of
the moon's orbit node,--the maximum correction for the true longitude
being only about 5d +/-.
[Illustration: Fig. 7]
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 [ARIES]EL the true longitude of the
pole of the lunar orbit, equal to the _true_ longitude of the ascending
node +/- 90d. VL is therefore the mean inclination +/- 2d 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 minutiae, 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.