HISTORY OP PHYSICAL ASTRONOMY.

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fixed planes of the satellites towards Jupiter ’s equator, counting from thefirst satellite, are only 7", V 3", 5' 2", 24' 33".

As the nodes of the satellites all regress upon their fixed planes, themutual positions of the orbits will be continually varying; and this cir-| cumstance, by occasioning a continual alteration of the mutual action ofi the satellites, will disturb the absolute positions of the orbits. Laplacel expressed the latitude of each satellite, relative to the plane of Jupiter ’sorbit, by means of five terms. The first of these depended on the positionof the fixed plane, with respect to the planet’s orbit; the second, on theinclination of the satellite towards its fixed plane; the other three weredetermined by the positions of the nodes of the disturbing satellites on! their fixed planes *.

I The analytical expression which Laplace obtained for the latitudei enabled him to explain the singular changes which astronomers had| remarked in the inclination of the fourth satellite. From 1680 till 1760,j the inclination appeared to have been stationary and equal to about 2°.4;

! since 1760 it had been sensibly increasing. Now, Laplace found from| theory, that in 1680 the inclination was equal to 2°.476; in 1620 itj amounted to 2°.448, and in 1760 to 2°.441. It reached its minimum: in 1756. Since that epoch it had been constantly increasing; whencomputed for 1800, by Laplace’s formula, it is found to amount to 2°.5791.“ It is curious,” says Laplace, “ to see thus emanating from analysis thosesingular phenomena which observation has partially disclosed; but which,resulting from the combination of many simple inequalities, are too com-plicated to allow the discovery of their laws by astronomers.”t

Since the motion of each satellite is determined by three differentialequations of the second order, the motions of the four satellites will bedetermined by twelve such differential equations. The integration of theseequations will, therefore, involve twenty-four arbitrary constants; thevalues of which must be derived from observation. Besides these twenty-four constants, there are seven others which it is necessary to determinebefore tables of the satellites can be constructed upon the basis of theirphysical theory. These are—the masses of the four satellites; theellipticity of Jupiter ; the inclination of his equator to his orbit; and theposition of the nodes of his equator. Five phenomena were selected byLaplace as best adapted for assigning the masses of the satellites and theellipticity of their primary. The first of these was the inequality inlongitude of the first satellite, extending over the period of 437.659 d ,which restores the eclipses of the three interior satellites in the sameorder. As this inequality depends on the action of the second satellite, itgave him the mass of that body with great accuracy. The similar ine-quality of the second satellite depends upon the combined actions of thefirst and third satellites ; it was the second datum employed by Laplace inthese researches. The third datum was the motion of the nodes of thesame satellite; a phenomenon which depends upon the action of the firstand third satellites, and upon the disturbing influence of Jupiter ’s ellip-

* The fixed planes of the satellites are not absolutely immoveable, since their positionsare determined by those of Jupiter ’s orbit and equator, both of which are continuallyvarying; the former from the action of the planets upon Jupiter , the latter from theaction of the sun and the satellites upon the redundant matter accumulated round hisequator. Laplace took into account the effects of both these changes in computing thelatitudes of the satellites.

An interesting account of the perturbations of Jupiter ’s satellites will he found in Airy’sTreatise on Gravitation.

t Miic. Cel., tome iv,, Pref. p. xiv.