might in reality be only one. He subsequently adopted this hypothesis as offering the best explanation of the phenomenon.

In 1781 he wrote to Lalande, stating that recent observations induced him to suppose there was only one inequality, with a period of about thirteen years. This inequality amounted to 74m between 1670 and 1720. From 1720 to 1760 it had diminished to 24m, and it had remained constant during the succeeding twenty years. This was not a happy modification of the original idea of two independent equations. Lagrange and Laplace have demonstrated, à priori, the existence of two distinct equations of the centre, in the motion of this satellite, and this remarkable result of pure theory has been confirmed in the most satisfactory manner by the laborious researches of Delambre.

Very little progress had been made by astronomers in the researches relative to the nodes of the satellites. In 1758 Maraldi invested this subject with a lively interest by the communication of a memoir to the Academy of Sciences, in which he announced that the nodes of the fourth satellite had a direct motion upon the plane of Jupiter's orbit. Newton, by considering the action of the sun upon the satellite, had found the motion to be retrograde, as in the case of the moon's nodes relative to the earth's orbit. Wargentin concurred with Maraldi in supposing that the motion was direct, and he fixed its annual value at 4' 15". This unexpected fact seemed to be at variance with the theory of gravitation, for Newton and his followers had shewn that the mean effect of a disturbing force was to occasion a retrograde motion of the nodes on the plane of the disturbing body. Lalande, however, shewed that the motion of the nodes might be direct upon one plane and retrograde upon another, and upon this ground he contended that, unless the principal disturbing force passed through the plane of Jupiter's orbit, the motion would not be necessarily retrograde. Now, in the present case, the third satellite exercises a much more powerful influence on the motion of the nodes than the sun does, and the same is true of the ellipticity of Jupiter. However, as neither the plane in which the satellite revolves, nor the plane of the planet's equator, coincides with the plane of the planet's orbit, it followed that the direct motion of the nodes on the latter plane could not be considered as invalidating the theory of gravitation.

The inclinations of the orbits presented great difficulties to astronomers, and formed the subject of much laborious research. We have seen that the elder Maraldi first discovered that the inclination of the second satellite is variable. Wargentin afterwards found that during fifteen years and a half the inclination continually increased, and that it then diminished by like degrees during other fifteen years and a halft. The whole period of variation was therefore equal to about thirty-one years. He fixed the extreme limits of the inclination at 3° 47′ and 1° 18'. Subsequently he made the greatest inclination 3° 46′, and the least 2° 46′. In 1765 Maraldi published a memoir, in which he shewed that the inequalities in the duration of eclipses could not be explained by the periodic change of the inclination. Proceeding on the supposition that the nodes were fixed, he calculated the inclination for two eclipses, ob served in 1714 and 1715, and he found that, during the eleven months embraced between them, it had increased to the extent of 20', or more than a fourth of the whole periodic variation. By a similar process he • Lalande, Traité d'Astronomie, tome iii. p. 145. Mém. Acad. Upsal, 1743.

found that, during thirteen months which elapsed between two eclipses, observed in 1750 and 1751, the inclination presented a diminution of 18' 20". These great changes occurred in such brief intervals, that it was impossible they could have proceeded from the periodic variation of the inclination. Maraldi finally discovered that the observations might be reconciled with the generally received theory of the inclination, by supposing the nodes to librate continually on each side of their mean place to the extent of 10° 13′ 48′′, the period of libration being equal to that which restored the same values of the inclination. Having compared this hypothesis with observation, he was gratified to find that a remarkable accordance generally prevailed between the results derived from both sources. Among 127 eclipses which he calculated, the difference between the observed and computed durations did not exceed 2m, except in one instance; and only 8 were found in which the difference exceeded 1m*.

Bailly shewed that the libration of the nodes proceeded from their retrograde motion on the orbit of the first satellite, which was in this case the principal disturbing body. He also remarked that the inclination was constant with respect to the orbit of this satellite, and that it was variable with respect to Jupiter's orbit, in consequence of the retrograde motion of the nodes upon the former orbit. This explained the coincidence of the period of libration with that which restored the inclination to the same value. When we consider the complicated character of the phenomenon investigated by Maraldi, his explanation of it must be regarded as one of the most ingenious conceptions which mere observation has ever suggested to the astronomer. Laplace has employed this libration of the nodes as one of the data from which he derived the masses of the satellites. Wargentin published tables of the second satellite for the Nautical Almanack for 1779; the most remarkable peculiarity of which was the libratory motion of the nodes, which he admits to have been first suggested by Maraldi.

The period in which the inclination of the third satellite passed through all its values was much longer than the corresponding period of the second satellite. Maraldi had found that during the current century it had been continually increasing. In 1763 he fixed it at 3° 25′ 41′′. This indicated an increase of 22' since the publication of Cassini's tables. In 1769 he discovered that the inclination was only 3° 23′ 33′′. Pursuing his researches, he found that the inclination reached its maximum in the years 1633 and 1765, and its minimum in 1699. Hence it followed that the inclination increased during 66 years, and then diminished during an equal space of time; the whole variation being consequently comprised within a period of 132 years. Maraldi also fixed the mean place of the nodes in 10s 13° 52', and estimated the extent of libration at 1° 32′ 24′′ †.

The inclination of the fourth satellite had long been considered invariable. We have seen that Bradley estimated it at 2° 42'. Wargentin, in 1750, made it 2° 39', and Maraldi, in 1758, made it 2° 36'. In 1781 Wargentin concluded from his researches that during a few preceding years the inclination had been slowly increasing. This opinion has been confirmed by the observations of subsequent astronomers, who have found that the increase of inclination has been going on down to the present day. Bailly, before becoming acquainted with the researches of Wargentin, made the following sagacious remark upon this subject in his History of Astronomy" The inclination of the fourth satellite has not hitherto

Mém. Acad. des Sciences, 1768. Ibid. 1769.

appeared to vary sensibly; but it will vary, for everything in the universe is subject to fixed laws; and the same circumstances always reproduce the same phenomena. We perceive merely that the period of the inclination must be very long and must extend to several centuries."*

This prediction has been verified by the physical researches of Laplace, who has found that the nodes have an annual retrograde motion of 4′32′′ upon a fixed plane, accomplishing a complete revolution in 532 years. It is this revolution of the nodes which occasions a variation of an equal period in the inclination of the satellite.

CHAPTER VIII.

Physical Theory of the Satellites.-Newton.-Euler.-Walmsley.-Bailly computes the Perturbations of the Satellites.-Researches of Lagrange. He obtains for each Satellite four Equations of the Centre and four Equations of Latitude. His mode of representing the Positions of the Orbits.-Inutility of his Theory in the Construction of Tables.-Laplace.-His Explanation of the constant Relations between the Epochs and Mean Motions of the three interior Satellites.-He completes the Physical Theory of the Satellites.-Delambre.—He calculates Tables on the Basis of Laplace's Theory. -He determines the Maximum Value of Aberration by means of the Eclipses of the first Satellite. Agreement of his Result with Bradley's.—Conclusions derivable from it. AFTER much laborious observation and research, the theory of the satellites was now sufficiently matured to form the basis of an explanation of their motions by the principle of universal gravitation. It is worthy of remark, that this is the order in which the various branches of astronomy have advanced towards their present high state of perfection. phenomena were first observed, and all the details relating to them carefully recorded. They were then submitted to a critical discussion, and, by a sagacious discrimination of their several peculiarities, they were grouped together under general laws. Finally these laws, although at first merely empiric, served the valuable purpose of suggesting the physical principles on which they depended; and when once this dependance was fully established, they henceforward assumed the more elevated character of laws of nature. This order of inquiry is especially manifest in the history of the lunar theory. A similar course has also been strongly recommended in our own day for the purpose of extending our knowledge of the Tides; and, indeed, it may be considered as offering the only means of ever conducting philosophers to a complete theory of that subject founded upon rigorous principles of geometry and physics.

The

If we only considered the disturbing action of the sun upon the satellites, the derangements in their motions would be in all respects analogous to those in the motion of the moon; and the analysis employed in the lunar theory would suffice for their complete investigation. In the present case, however, the problem is much more complicated. Each satellite is

disturbed, not only by the sun, but by the other three satellites in course of every synodic revolution round the central body. Nor are these the only sources of complication. If Jupiter were a perfect sphere, his attraction would be the same, both in quantity and direction, as if his whole mass were collected at his centre; and the question relative to his action upon each

Bailly, Hist. Ast. Mod., tome iii. p. 183.

satellite would be reduced to the simple consideration of a material particle attracting the body at that point. This, however, is not the real case of nature; for observation shews that the figure of the planet is that of an oblate spheroid, whose axes are to each other nearly as 13 to 14. This circumstance causes the law of his attraction to deviate from the inverse square of the distance, and hence originates a disturbing force which powerfully deranges the motions of the satellites.

Newton, in the third book of the Principia, considers the disturbing action of the sun upon Jupiter's satellites, and attempts to determine the inequalities of their motions by the principles of the lunar theory. In this manner he found that the nodes of the fourth satellite had a retrograde motion upon the plane of Jupiter's orbit, the annual quantity of which amounted to 5'*. We have seen that this result was subsequently contradicted by observation; the actual motion having been discovered by astronomers to be direct. The phenomenon in question does not, in fact, depend so much upon the sun as upon the third satellite and the ellipticity of Jupiter; causes of disturbance which were not taken into account by Newton.

Euler, in 1748, first remarked that the spheroidal figure of Jupiter would occasion an irregularity in the law of his attraction t. Walmsley, in 1758, shewed that the disturbance hence arising would produce a motion of the nodes and apsides of each satellite. In 1763 Euler communicated a memoir to the Academy of Berlin, in which he examined the perturbations of a satellite revolving round a planet of a spheroidal figure. He shewed that when the satellite revolved in the plane of the planet's equator the action of the protuberant matter generally occasioned a progressive motion of the apsides. As the orbit of the satellite became more inclined to this plane, the motion of the apsides continually diminished, and it ceased altogether when the angle of inclination was equal to 54° 44'. From this position the motion of the apsides was regressive, and it continued to increase until the orbit of the satellite was perpendicular to the plane of the equator.

Bailly S, about the same time, employed Clairaut's theory of the moon in researches on the perturbations of the satellites. He discovered, by a simple analysis, that the mutual attraction of the three interior satellites occasioned those inequalities in their motions which produced a regular return of their eclipses at the end of 437 days. We have seen that Bradley was the first astronomer who threw out a suspicion of this fact. These inequalities are precisely analogous to the lunar variation, the only difference being, that the disturbing body is in each case one of the satel lites themselves, and not the sun. In the theory of the first satellite the principal disturbing body is the second, for the exterior satellites are too remote to exercise any sensible influence, and the effect of Jupiter's ellipticity is equally inappreciable, because the orbit of the satellite is situated in the plane of his equator, and at the same time does not possess any eccentricity. It is clear, then, that by comparing the coefficient of the equation furnished by theory with the magnitude of the inequality,

Principia, lib. iii. prop. 23.

Newton, in the same proposition, makes the inequality of the fourth satellite depending on the disturbing action of the sun, and, similar to the lunar variations, equal to 5′ 12′′.

+ Recherches des Inégalités de Jupiter et de Saturne. Prix de l'Académie, tome vii. Phil. Trans., 1758.

Born at Paris in 1736; perished by the guillotine in 1793.

as assigned by observation, the mass of the second satellite may be readily determined. In this manner Bailly found it to be equal to 0.0000211, Jupiter's mass being supposed equal to unity. This was a tolerable approximation to the true value. Laplace makes the mass equal to 0.0000232. The inequality of the second satellite is essentially a more complex phenomenon than that of the first, for it depends on the combined action of the first and third satellites. In form, however, the two inequalities are precisely similar, the effects of the disturbing bodies in the theory of the second satellite being blended together so as to form one great inequa lity, governed by the same law, and extending over the same period as the inequality of the first satellite. This singular coincidence derives its origin from two remarkable relations, connecting together the mean longitudes and mean motions of the three interior satellites. Bailly found that the equation of sensible magnitude, depending on the action of the first satellite, is expressed by the sine of the difference between the mean longitudes of the first and second, and that the equation of a similar nature, depending on the action of the third satellite, is expressed by the sine of twice the difference between the mean longitudes of the third and second *. Now, observation shewed that these two arcs differed from 180° by only a very small quantity. Wargentin's tables, in fact, suppose the difference to be equal only to 30' at the commencement of the year 1760. The arcs being therefore nearly supplementary to each other, it followed that their sines were equal, and hence Bailly was enabled to combine the two equations together, by merely adding their coefficients and retaining either of the arguments. It is in consequence of this union of the effects of the two disturbing satellites that the inequality of the second satellite exceeds so much the analogous inequalities in longitude of the first and third.

The derangements produced by the two disturbing satellites being thus confounded together, it was impossible to pronounce how much of the resulting inequality was due to each satellite; and hence, in order to determine the masses of those bodies, another independent datum, derived from observation, was indispensable. Bailly selected for this purpose the motion of the nodes of the second satellite. This phenomenon depends on the action of the first and third satellites, and also upon the disturbance occasioned by the oblate figure of Jupiter. Having formed an ingenious supposition respecting the density of the planet, Bailly computed the effect of his oblateness in disturbing the place of the nodes, and then, subducting the result from the observed motion, he obtained the quantity due to the action of the two satellites. Combining this datum with the one assigned by the inequality in longitude, he determined the masses of • In both cases there are terms depending on the two arguments mentioned in the text, but, when the first satellite is the disturbing body, the term having for its argument the difference between the mean longitudes greatly exceeds all the others; and, on the other hand, when the third satellite is the disturbing body, the most considerable term is that depending on twice the difference between the mean longitudes. The predominance of these terms arises from the fact, that twice the mean motion of the second satellite is very nearly equal to the mean motion of the first, and twice the mean motion of the third satellite is very nearly equal to the mean motion of the second. The circumstance is, indeed, exactly similar to that which gives rise to the long inequality in the theory of Jupiter and Saturn, or to the analogous inequality in the theory of the Earth and Venus. The inequalities of the satellites differ, however, from those just cited, in being independent of the eccentricities. Their investigation will be found in Woodhouse's Physical Astronomy, chapter xx. For the more intricate parts of the theory of the satellites, see the Mécanique Celeste, liv. viii.; also Mrs. Somerville's Mechanism of the Heavens, book iv.