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position of the lunar orbit, since it is not in that plane. He determined the effect of this force, by supposing AB, fig. 107, to be the arc described by the moon in an instant; then ACB is the plane of the orbit during that time; in the next instant, the difference of the two forces causes the moon to describe the small arc BD in a D different plane; then if BD represent the difference of the forces, and if AB be the velocity of the moon in the first instant, the diagonal BD will be the direction of the velocity in the second instant; and ACD will be the position of the orbit. Newton deduced the horary and mean motion of the nodes, their principal variation, and the inequalities in latitude, from these considerations. La Place considered the theory of the moon as the most profound and ingenious part of the Principia.
THEORY OF JUPITER'S SATELLITES.
798. JUPITER is attended by four satellites, which were discovered by Galileo on the 1st of June, 1610; their orbits are nearly in the plane of Jupiter's equator, and they exhibit all the phenomena of the solar system, on a small scale and in short periods. The eclipses of these satellites afford the easiest method of ascertaining terrestrial longitudes; and the frequency of the occurrence of an eclipse renders the theory of their motions nearly as important to the geographer as that of the moon.
799. The orbits of the two first satellites are circular, subject only to such eccentricities as arise from the disturbing forces; the third and fourth satellites have elliptical orbits; the eccentricity however is so small, that their elliptical motion is determined along with those perturbations that depend on the eccentricities of the orbits.
800. Although Jupiter's satellites might be regarded as an epitome of the solar system, they nevertheless require a new investigation, on account of the nearly commensurable ratios in the mean motions of the three first satellites, the action of the sun, the ellipticity of Jupiter's spheroid, and the displacement of his orbit by the action of the planets.
801. It appears, from observation, that the mean motion of the first satellite is nearly equal to twice that of the second; and that the mean motion of the second is nearly equal to twice that of the third; whence the mean motion of the first, minus three times that of the second, plus twice that of the third, is zero; but the last ratio is so exact, that from the earliest observations it has always been zero.
It is also found that, from the time of the discovery of the satellites, the mean longitude of the first, minus three times that of the second, plus twice that of the third, is equal to 180°: and it will be shown, in the theory of these bodies, that even if these ratios had not been exact in the origin of their motions, their mutual attractions would have made them so. They are the cause of the principal inequalities in the longitude of the satellites; and as they exist also in their synodic motions, they have a great influence on the times of their eclipses, and indeed on their whole theory.
802. The prominent matter at Jupiter's equator, together with the action of the satellites themselves, causes a direct motion in the apsides, which changes the relative position of the orbits, and alters the attractive force of the satellites; consequently each satellite has virtually four equations of the centre, or rather, that part of the longitude of each satellite that depends on the eccentricity, consists of four principal terms; one that arises from the true ellipticity of its own orbit, and three others, depending on the positions of the apsides of the other three orbits. Inequalities perfectly similar to these are produced in the radii vectores by the same cause, consisting of the same number of terms, and depending on the same quantities.
803. Astronomers imagined that the orbits of the satellites had a constant inclination to the plane of Jupiter's equator; however, they have not always the same inclination, either to the plane of his equator or orbit, but to certain imaginary fixed planes passing between these, and also through their intersection.
Let NJN' be the orbit of Jupiter, NQN' the plane of his equator extended so as to cut his orbit in NN'; then, if NMN' be the orbit of a satellite, it will always preserve very nearly the same inclination to a fixed plane NFN', passing between the planes NQN' and NJN', and through the line of their nodes. But although the orbit of the satellite preserves nearly the
same inclination to NFN', its nodes have a retrograde motion on that plane. The plane FN itself is not absolutely fixed, but moves slowly with the equator and orbit of Jupiter. Each satel
lite has a different fixed plane, which is less inclined to the plane of Jupiter's equator the nearer the satellite is to the planet, evidently arising from the attraction of the protuberance at Jupiter's equator, which retains the satellites nearly in the plane of the equator; furnishing another proof of the mutual attraction of the particles of
804. The equatorial matter of Jupiter's spheroid causes a retrograde motion in the nodes of the orbits of the satellites; which alters their mutual attraction, by changing the relative position of their planes, so that the latitude of any one satellite not only depends on the position of the node of its own orbit, but on the nodes of the other three; and as the position of Jupiter's equator is perpetually varying, in consequence of the action of the sun and satellites, the latitude of these bodies varies also with the inclination of Jupiter's equator on his orbit, and the position of its nodes. Thus, the principal inequalities of the satellites arise from the compression of Jupiter's spheroid, and from the direct and indirect action of the sun and satellites themselves.
805. The secular variation in the form and position of Jupiter's orbit is the cause also of secular variations in the motions of the satellites, similar to those in the motions of the moon occasioned by the variation in the eccentricity and position of the earth's orbit.
806. The position of the orbit of a satellite may be known by supposing five planes, of which FN, passing between JN and QN, the planes of Jupiter's orbit and equator, always retains very nearly the same inclination to them. The second plane An moves uniformly on FN, retaining nearly the same inclination on it. The
third Bn' moves in the same manner on An; the fourth Cn" moves similarly on Bn'; and the fifth Mn", which has the same kind of motion on Cn", is the orbit of the satellite. The motion of the nodes are retrograde, and each satellite has a set of planes peculiar to itself. In conformity with this, the latitude of a satellite above the variable orbit of Jupiter, is expressed by five terms; the first of which is relative to the displacement of the orbit and equator of
Jupiter; the second is relative to the inclination of the orbit of the satellite on its fixed plane; and the other three terms depend on the position and motion of the nodes of the other three orbits. The inequalities which have small divisions, arising from the configuration of the bodies, are insensible in latitude, with the exception of those produced by the sun, which modify the preceding quantities.
807. For the solution of the problem of the satellites, the data that must be determined by observation for a given epoch, are, the compression of Jupiter's spheroid, the inclination of his equator on his orbit, the longitude of its nodes, the eccentricity of his orbit, its position, and its secular variations; the masses of the four satellites, their mean distances, periodic times, the eccentricities and inclinations of their orbits, together with the longitude of their apsides and nodes. The masses of the satellites and the compression of Jupiter are determined from the inequalities of the satellites themselves.
808. The orbits of the four satellites may be regarded as circular, because the eccentricity of the third, and even the fourth, is so small, that their equations of the centre will be determined with the perturbations depending on the eccentricities and inclinations. Thus, with regard to the two first, and nearly for the other two, the true longitude is the sum of the mean longitude and perturbations; and the radius vector will be found by adding the perturbations to the mean distance.
809. A satellite m is troubled by the other three, by the sun, and by the excess of matter at Jupiter's equator. The problem however will be limited to the action of the sun, of Jupiter's spheroid, and of one satellite; the resulting equations will be general, from whence the action of each body may be computed separately, and the sum will be the effect of the whole.
810. Let m and m, be the masses of any two satellites, x, y, z, x', y', z', their rectangular co-ordinates referred to the centre of gravity of Jupiter, supposed to be at rest; r, r' their radii vectores; then the disturbing action of m, on m is
= R ;
consequently the sign of R must be changed in equations (155) and (156), since it is assumed to be negative in this case.