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.