to the plane of the planet’s equator, there is a want of symmetry, and the action of the protuberant matter tends to make the nodes regress by pulling the satellites above or below the planes of their orbits; an action which is so great on the interior satellites, that the motions of their nodes are nearly the same as if no other disturbing force existed. The orbits of the satellites do not retain a permanent inclination, either to the plane of Jupiter’s equator, or to that of his orbit, but to certain planes passing between the two, and through their intersection. These have a greater inclination to his equator the farther the satellite is removed, owing to the influence of Jupiter’s compression; and they have a slow motion corresponding to secular variations in the planes of Jupiter’s orbit and equator. The satellites are not only subject to periodic and secular inequalities from their mutual attraction, similar to those which affect the motions and orbits of the planets, but also to others peculiar to themselves. Of the periodic inequalities arising from their mutual attraction the most remarkable take place in the angular motions (N. 89) of the three nearest to Jupiter, the second of which receives from the first a perturbation similar to that which it produces in the third; and it experiences from the third a perturbation similar to that which it communicates to the first. In the eclipses these two inequalities are combined into one, whose period is 437·659 days. The variations peculiar to the satellites arise from the secular inequalities occasioned by the action of the planets in the form and position of Jupiter’s orbit, and from the displacement of his equator. It is obvious that whatever alters the relative positions of the sun, Jupiter, and his satellites, must occasion a change in the directions and intensities of the forces, which will affect the motions and orbits of the satellites. For this reason the secular variations in the excentricity of Jupiter’s orbit occasion secular inequalities in the mean motions of the satellites, and in the motions of the nodes and apsides of their orbits. The displacement of the orbit of Jupiter, and the variation in the position of his equator, also affect these small bodies (N. 90). The plane of Jupiter’s equator is inclined to the plane of his orbit at an angle of 3° 5ʹ 30ʺ, so that the action of the sun and of the satellites themselves produces a nutation and precession (N. 91) in his equator, precisely similar to that which takes place in the rotation of the earth, from the action of the sun and moon. Hence the protuberant matter at Jupiter’s equator is continually changing its position with regard to the satellites, and produces corresponding mutations in their motions. And, as the cause must be proportional to the effect, these inequalities afford the means, not only of ascertaining the compression of Jupiter’s spheroid, but they prove that his mass is not homogeneous.