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HISTOBY OF PHYSICAL ASTBONOMY.

depending upon the action of the fourth, the orbit of which is considerablyeccentric. The combination of this equation with the satellite’s own equationof the centre gives rise to a single equation of variable magnitude, which im-parts a somewhat complicated character to the motion of the satellite, andrenders it difficult to trace the respective sources of each inequality by theaid of mere observation. Laplace found from theory that the lower apsidesof the two satellites coincided in the year 1082; and, in consequence, the twoequations combined together into one equation equal to their sum, andamounting to 796".411 of space. In 1777 the lower apsis of the thirdsatellite had advanced 180° before that of the fourth, and the resulting equa-tion was equal only to the difference of the two elementary equations. In thisposition it therefore only amounted to 607".651. These results are exactlyconformable to those which we have seen that Wargentin and otherastronomers had previously derived from observation. In his researches on.the perturbations in latitude, Laplace gave a strong proof of his sagacity bytaking into account the effect of the inclination of Jupiter ’s equator to bisorbit. We have already remarked that this important element of disturb-ance was entirely omitted by Lagrange.

The general tendency of Jupiter ’s ellipticity is to draw the satellites intothe plane of his equator. This is clearly seen in the actual positions ofthe orbits, which in each case deviate to a greater extent from the plane ofthe equator, according as the satellite is more remote from its primary.The inclinations of the orbits had occasioned much trouble to astronomers,chiefly in consequence of the difficulty of finding a plane of reference withwhich they might be connected by fixed relations. Laplace discovered thatthe orbit of each satellite revolved with a constant inclination upon a fixedplane contained between the planes of Jupiter ’s equator and orbit, and passingthrough their common intersection. If there had been no other disturbingbody than the sun, the fixed plane of each satellite would have been the planeof Jupiter’s orbit; also, if the protuberant matter around Jupiter ’s equatoralone disturbed the satellite, the fixed plane would have coincided with theplane of his equator ; and a similar result would follow if any of the otherthree satellites were the sole disturbing body. It is clear, then, that theactual position of the fixed plane will in each case be determined by areference to the opposing tendencies of the five disturbing forces, and itwill manifestly bold an intermediate place between the planes of Jupiter ’sorbit and equator, which are the two extreme planes in which thesedisturbing forces act. This conclusion agrees with what we have mentionedabove, relative to Laplace’s determination of these planes. The fixedplane of each satellite might, therefore, be considered as the resultantplane of the disturbing forces; and the motion of the nodes upon it wasaccordingly found by Laplace to be retrograde, conformably with thegeneral effect of a disturbing force acting continually in the plane. Thisresult offered a satisfactory explanation of the curious movements of thenodes on the plane of Jupiter ’s orbit; phenomena which, when first recog-nised by astronomers, appeared to be irreconcileable with the principles ofthe Newtonian theory, in consequence of the plane of Jupiter ’s orbithaving been erroneously assumed to be the plane of the disturbing force.

The disturbing force which exercises most influence in determining thepositions of the fixed planes, is that arising from the ellipticity of Jupiter ;and this force is obviously more effective relatively to the sun’s action,according as the satellite is more remote from its primary. The inclina-tion of Jupiter 's equator to his orbit is 3° 5'; but the inclinations of the