Next, consider a tangential accelerating force. Here the immediate effect is to increase the velocity at any point of the orbit, and therefore to make it correspond to a larger orbit, and, consequently, a longer periodic time. Conversely, a retarding force, such as the resistance of a medium, diminishes the velocity at each point, and thus makes the motion correspond to that iu an ellipse with a less major axis, and therefore with a diminished periodic time. This singular result, that the periodic time of a body is diminixIced by resistance, is realized in the case of Eneke's comet, and this observed effect furnishes one of the most convincing proofs of the existence of a resisting medium it) inteiplanetary space.
Again. the effect of a disturbing force continually directed toward the plane of the ecliptic, is to make the node regrede. Thus, if N'N represent the ecliptic, NM a portion of the orbit, the tendency of the disturbing force at 31 is to make 3IQ the new orbit, and therefore N' the node. Thus the node regredes, and the inclination of the orbit to the ecliptic is diminished, when the planet has just passed the ascending node. In the second figure, let 31, be a position of the planet near the descending node N,. The effect of the disturbing force is to alter the orbit to MN,'. Thus, again, the node regredes, but the inclination is increased. If NN' and N,N,' in these figures represent the earth's equator, the above rough sketch applies exactly to the case of the moon as disturbed by the oblateness of the earth. The tion of the moon on the earth gives rise to the cession of the equinoxes (q.v.).
By processes of this nature, Newton subjected the variation of the elements of the moon's orbit to calculation, and obtained the complete explanation of some of the most important of the lunar inequalities. See MooN. Others of them—for instance, the rate of progression of the apse—cannot be deduced with any accuracy by these rough investigations, but tax, in some cases, the utmost resources of analysis. Newton's calculation of the rate of the moon's apse was only about half the observed value; and Clairaut was on the point of publishing a pamphlet, in which a new form was suggested for the ;au- of gravitation, in order to account for the deficiency of this estimate; when he found, by carrying his analysis further, that the expression sought is obtainable in the form of a slowly converging series, of which the second term is nearly a large as the first. The error of the modern lunar tables, founded almost entirely on analysis, with the necessary introduc tion of a few data from observation, rarely amounts to a second of arc; and the moon's place is predicted four years beforehand, in the Nautical Almanac, with a degree of precision which no mere observer could attain even from one day to the next.. This is the true proof,
not only of the law of gravitation, but of the laws of motion (q.v.), upon which, of course, the analytical in vestigation is based.
With respect to the mutual perturbations of the planets, we may merely mention that they are divisible into two classes, called periodic and secular. The former depend upon the configurations of the system—such, for instance, is the diminution of the inclination of the moon's orbit, 4fter passing the ascending node on the earth's equator, already mentioned, or its increase as the moon comes to the descending node. The secular per turbations depend upon the period in which a complete series of such alternations have been gone through, and have, in the case of the planets, complete cycles measured by hundreds of years.
A very curious kind of perturbation is seen in the indireet action of the planets on the moon. There is a secular change of the eccentricity of the earth's orbit, due to planetary action, and this brings the sun, on the average, nearer to the earth and moon for a long period of years, then for an equal period takes it further off. One of the effects of the sun's force being, as we have seen, to diminish, on the whole, the moon's gravity toward earth, this diminution will vary in the same period as the eccentricity of the earth's orbit; and therefore the moon's mean motion will be alternately accelerated and retarded, each process occupying an immense period.
With special reference to the planetary motions, we may notice that the major axis of each planetary orbit is free from all secular variations; and those affecting the inclina tion and eccentricity are confined within small limits, and ultimately compensate them selves. These facts, which have been clearly and beautifully demonstrated by Laplace and Lagrange, assure the stability of the planetary orbits, if we neglect the effects of resistance clue to the interplanetary matter; which, however, must, in the long run, bring all the bodies of the system into collision with the sun, and finally stop the rotation of the sun itself, Newton commenced the investigation of perturbations by considering those of the moon ; Euler followed with a calculation of Saturn's inequalities; while Clairaut, D'Alembert, and others successively gave those of the other planets.
Every one knows that it was by observing the pertUrbations of Uranus, and thence discovermg the direction of the disturbing force, that Adams and Leverrier were to their great and simultaneous discovery of the planet Neptune,