When one of the four Galilean satellites is in transit across the disk of Jupiter it can generally be seen projected on the face of the planet. It is commonly brighter than Jupiter when it first enters upon the limb but sometimes darker near the centre of the disk. This is owing to the fact that the planet is much darker at the limb. During these transits the shadow of the satellites can also be seen projected on the planet as a dark point. The theories of the motion of these bodies form one of the more interesting problems of celestial mechanics. Owing to the great ellipticity of Jupiter, growing out of his rapid rotation, the influence of this ellipticity upon the motions of the five inner satellites is much greater than that of the sun, or of the satellites on each other. The inclination of the orbits to the equator of Jupiter is quite small and almost constant, and the motion of each node is nearly uniform around the plane of the planet's equator. The most marked feature of these bodies is a relation between the mean longitudes of Satellites I., II. and III. The mean longitude of I. plus twice that of III. minus three times that of II. is constantly near to 180°. It follows that the same relations subsist among the mean motions. The cause of this was pointed out by Laplace. If we put L1 L2 and L3 for the mean longitudes, and define an angle U as follows:-- U = L1 - 3 L2 + 2 L3. it was shown mathematically by Laplace that if the longitudes and mean motions were such that the angle U differed a little from 180°, there was a minute residual force arising from the mutual actions of the several bodies tending to bring this angle towards the value 180°. Consequently, if the mean motions were such that this angle increased only with great slowness, it would after a certain period tend back toward the value 180°, and then beyond it, exactly as a pendulum drawn out of the perpendicular oscillates towards and beyond it. Thus an oscillation would be engendered in virtue of which the angle would oscillate very slowly on each side of the central value. Computation of the mean longitude from observations has indicated that the angle does differ from 180°, but it is not certain whether this deviation is greater than the possible result of the errors of observation. However this may be, the existence of the libration, and its period if it does exist, are still unknown. The following are the principal elements of the orbits of the five inner satellites, arranged in the order of distance from Jupiter. The mean longitudes are for 1891, 20th of October, G.M.T., and are referred to the equinox of the epoch, 1891, 2nd of October:--