It thus appears that by far the Largest part of the effect of the natation arises front the moon, and depends not upon the place of the planet, hut upon that of its node. 'Ilia node Plooxj performs a complete regressive revolution in about eighteen years and a half, in which time the main effect of notation goes through all its changes Bradley, the discoverer of the mutation, found out and assigned the law of its largest term ; blue remaining ones are due to the theory of gravitation. It took him twenty years of observation, somewhat more tlum the whole period of the change, to detect time law of the anomaly whose existence he became aensible of iintnediately after his discovery of aberration had cleared away the largest part of the then unexplained motions of the eters. The discovery was completed and published at the end of 1747. In 1749 appeared the 'Recherches Bur la Precession, &c. of D'Alembert, in which the phenomenon was shown ..to be the necessary consequence of the moon's attraction upon the earth. Newton had already, in the Principia; given the general explanation of the subject, and had even foretold, without assigning magnitudes, the existence of those terms of nutation which depend upon twice the true longitudes of the sun and moon ; but the most important terms, those depending on the moon's node, appear to have been altogether unsuspected by him.
We now come to such a physical explanation of the cause of pre cession and nutation as can be given without mathematical analysis. On looking at the motion of the equator arising from precession and nutation, we see that it precisely resembles in character some of the alterations which take place in a planet's orbit, the precession answering to the regression of the nodes, the equation of the equinoxes to the variation of that regression, and the remaining part of the nutation, or the variation of the obliquity, to the variation of the inclination to the ecliptic. It was soon seen by Newton, that on the supposition of the mutual attraction of all the particles of matter, the action of the heavenly bodies on the protuberant parts of the earth must produce exactly that sort of effect on the motion of the equator which the disturbing force of the sun, for instance, produces on the moon. He thus explains, firstly, the precession ; secondly, that part of the natation of the inclination which depends upon twice the longitude of ale disturbing body. This explanation (prop. 66, corol laries 18-22) is substantially as follows :— If a sphere in rotation be attracted by another body, the axis of rotation must remain unaltered : for since a plane drawn through any attracting point and the centre of the attracted sphere cuts the sphere into two perfectly similar halves, there is no effect upon the rotation (or tendency to an effect) arising from the attraction upon one half of the sphere which is not destroyed by the tendency to the exactly opposite effect arising from the attraction .upon the other half. if then the earth were a perfect sphere, whatever motion of translation the whole sphere might receive, the axis would always remain parallel to its first position, and there would be neither precession nor nutation.
Again, let the earth be a solid of revolution, protuberant for example at the equator, as is the case, and let an attracting point be situated in the plane of the equator ; the symmetry just alluded Ito still exists, and the result is the same. But if an attracting point be not situated in the plane of the equator, the plane passing through the attracting point and the centre divides the spheroid into parts which, though equal, are no longer similarly situated with respect to the attracting point. The alteration of the axis which would take place if one half only were attracted, is no longer counterbalanced by the attraction on the other half : the direction of the axis is therefore continually changed.
To get a specific idea of the nature of the change, first suppose the spherical part of the earth only to exist, the protuberance being removed; and the solidity of the sphere still remaining. let all its matter be supposed to be removed to the centre at c (Pig. 2). The diagram shows the spheroid of the earth, distinguishing the inscribed sphere from the protuberant part ; the solidity of both parts _is supposed to remain, but the matter of the internal sphere is removed to e, that of the protuberant part is not yet introduced : at is the attracting body, and the plane of its orbit is given, while the directions of the earth's rotation and of It's orbital motion are denoted by arrows. At present at pmluces no effect on the rotation ; now let a small mass of matter be affixed to the equator at z, which will therefore move round the mass c in and with the equator. The consequence will be [GitsvirsrioN], that the node of this orbit (the equinox A) will regress, or move in the direction opposite to that of the arrows, while the inclination of the orbit will alternately increase and diminish, being greatest when the line oat imams through one of the equinoxes. If we put such satellites all round the equator, the effect will not be altered in quality, but increased in magnitude ; and if we fill up all the protuberant part of the spheroid, the effect will still be of the same sort, though further increased in magnitude. The effect of the parts of the protuberance nearer to the pole is, for a given mass, less than that of the parts near the equator. Finally, if we restore the mass of the internal sphere to its proper place, the effect will be less than before ; for since no motion of the protuberant part can take place without one of the whole sphere, and since rotation is more difficult to produce, the greater the distance of the masses moved from the axis, the distribution of the mass at a over all parts of the sphere will render at less efficient in the alteration of the direction of the axis, Thus it appears that the phenomena of precession and nutation may arise from the consideration of the protuberant part of the spheroid as a fixed satellite to the internal part ; but the proof that the precession and nutation do so arise consists in taking a strict mathematical process, investigating the precession and nutation in quantity as well as quality, and showing that the results agree with those of obser vation.