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but constant inclinations; however, his equations of the latitude were incomplete, from the error of assuming Jupiter's equator to be on the plane of his orbit. It was reserved for La Place to perfect this important theory, by including in these equations the inclination of Jupiter's equator, the effects of his nutation, precision, and the displacement of his orbit, and also by the discovery of the four fixed planes, of the libration, and of the law in the mean longitudes, discoveries that rank high among the many elegant monuments of genius displayed in his system of the world. The perfect harmony of these laws with observation, affords one of the numerous proofs of the universal influence of gravitation. They are independent of secular inequalities, and of the resistance of a rare medium in space, since such resistance would only cause secular inequalities so modified by the mutual attraction of the satellites, that the secular equation of the first, minus three times that of the second, plus twice that of the third, would always be zero; therefore the inequalities in the return of the eclipses, whose period is 437 days, will always be the same.
953. The libration by which the three first satellites balance each other in space, is analogous to a pendulum performing an oscillation in 1135 days. It influences all the secular variations of the satellites, although only perceptible at the present time in the inequality depending on the equation of the centre of Jupiter; and as the observations of Sir William Herschel shew that the periods of the rotation of the satellites are identical with the times of their revolutions, the attraction of Jupiter affects both with the same secular inequalities.
954. Thus Jupiter's three first satellites constitute a system of bodies mutually connected by the inequalities and relations mentioned, which their reciprocal action will ever maintain if the shock of some foreign cause does not derange their motion and relative position: as, for instance, if a comet passing through the system, as that of 1770 appears to have done, should come in collision with one of its bodies. That such collisions have occurred since the origin of the planetary system, is probable: the shock of a comet, whose mass only equalled the one hundred thousandth part of that of the earth, would suffice to render the libration of the satellites sensible; but
since all the pains bestowed by Delambre upon the subject did not enable him to detect this, it may be concluded that the masses of any comets which may have impinged upon one of the three satellites nearest to Jupiter must have been extremely small, which corresponds with what we have already had occasion to observe on the tenuity of the masses of the comets, and their hitherto imperceptible influence on the motions of the solar system.
955. To complete the theory, thirty-one unknown quantities remained to be derived from observation, all of which Delambre determined from 6000 eclipses, and with these data he computed tables of the motions of the satellites from La Place's formulæ, subsequently brought to great perfection by Mr. Bouvard.
The Satellites of Saturn.
956. Saturn is surrounded by a ring, and seven satellites revolve from west to east round him, but their distance from the earth is so great that they are only discernible by the aid of very powerful telescopes, and consequently their eclipses have not been determined, their mean distances and periodic times alone have been ascertained with sufficient accuracy to prove that Kepler's third law extends to them. If 8".1 the apparent equatorial semidiameter of Saturn in his mean distance from the sun be assumed as unity, the mean distances and periodic times of the seven satellites are,
The masses of the satellites and rings and the compression of Saturn being unknown, their perturbations cannot be determined. The orbits of the six interior satellites remain nearly in the plane of Saturn's equator, owing to his compression, and the reciprocal attraction of the bodies.
The orbit of the seventh satellite has a motion nearly uniform on a fixed plane passing between the orbit and equator of that planet, inclined to that plane at an angle of 15°.264. The nodes have a retrograde annual motion of 304".6; the fixed plane maintains a constant inclination of 21°.6 to Saturn's equator, but the approximation must be imperfect that results from data so uncertain.
957. The action of Saturn on account of his compression, retains the rings and the orbits of the six first satellites in the plane of his equator. The action of the sun constantly tends to make them. deviate from it; but as this action increases very rapidly, and nearly as the 5th power of the radius of the orbit of the satellite, it is sensible in the seventh only. This is also the reason why the orbits of Jupiter's satellites are more inclined in proportion to their greater distance from their primary, because the attraction of his equatorial matter decreases rapidly, while that of the sun increases. When the seventh satellite is east of the planet, it is scarcely perceptible from the faintness of its light, which must rise from spots on the hemisphere presented to us. Now, in order to exhibit always the same appearance like the moon and satellites of Jupiter, it must revolve on its axis in a time equal to that in which it revolves round its primary. Thus the equality of the time of rotation to that of revolution seems to be a general law in the motion of the satellites.
The compression of Saturn must be considerable, its revolution being performed in 11h 42′ 43′′, nearly the same with that of Jupiter.
Satellites of Uranus.
958. The slow motion of Uranus in its orbit shows it to be on the confines of the solar system. Its distance is so vast that its apparent diameter is but 3.9, its satellites are therefore only within the scope of instruments of very high powers; Sir William Herschel discovered six revolving in circular orbits nearly perpendicular to the plane of the ecliptic. Taking the semidiameter of the planet for unity, their mean distances and periodic times are
subject, therefore, to the third law of Kepler. The compression of their primary and their reciprocal attraction retains their orbits in the plane of the planet's equator.
Arcs, circular, convertible into time
introduced by integration into series of perturbations 299, 312
Areas, principle of, in a system of bodies
in a rotating solid
Areas, variation of, a test of disturbing forces
sum of, zero on two of the co-ordinate planes and a maximum on