here, 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.

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.

[Illustration: Fig. 8]

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

similarity Ct : Tt :: Cm : Mm
but Tt : Mm :: Moon's mass : Earth's mass.
That is Tt : 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 ar