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 2° 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 5° ±.
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 ± 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.