fixed planes of the satellites towards Jupiter's equator, counting from the first satellite, are only 7", 1′ 3′′, 5′ 2′′, 24′ 33′′.

As the nodes of the satellites all regress upon their fixed planes, the mutual positions of the orbits will be continually varying; and this circumstance, by occasioning a continual alteration of the mutual action of the satellites, will disturb the absolute positions of the orbits. Laplace expressed the latitude of each satellite, relative to the plane of Jupiter's orbit, by means of five terms. The first of these depended on the position of the fixed plane, with respect to the planet's orbit; the second, on the inclination of the satellite towards its fixed plane; the other three were determined by the positions of the nodes of the disturbing satellites on their fixed planes *.

The analytical expression which Laplace obtained for the latitude enabled him to explain the singular changes which astronomers had remarked in the inclination of the fourth satellite. From 1680 till 1760, the inclination appeared to have been stationary and equal to about 2o.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 it amounted to 2o.448, and in 1760 to 2o.441. It reached its minimum in 1756. Since that epoch it had been constantly increasing; when computed 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 those singular phenomena which observation has partially disclosed; but which, resulting from the combination of many simple inequalities, are too complicated to allow the discovery of their laws by astronomers."†

Since the motion of each satellite is determined by three differential equations of the second order, the motions of the four satellites will be determined by twelve such differential equations. The integration of these equations will, therefore, involve twenty-four arbitrary constants; the values of which must be derived from observation. Besides these twentyfour constants, there are seven others which it is necessary to determine before tables of the satellites can be constructed upon the basis of their physical theory. These are-the masses of the four satellites; the ellipticity of Jupiter; the inclination of his equator to his orbit; and the position of the nodes of his equator. Five phenomena were selected by Laplace as best adapted for assigning the masses of the satellites and the ellipticity of their primary. The first of these was the inequality in longitude of the first satellite, extending over the period of 437.6594, which restores the eclipses of the three interior satellites in the same order. As this inequality depends on the action of the second satellite, it gave him the mass of that body with great accuracy. The similar inequality of the second satellite depends upon the combined actions of the first and third satellites; it was the second datum employed by Laplace in these researches. The third datum was the motion of the nodes of the same satellite; a phenomenon which depends upon the action of the first and third satellites, and upon the disturbing influence of Jupiter's ellip* The fixed planes of the satellites are not absolutely immoveable, since their positions are determined by those of Jupiter's orbit and equator, both of which are continually varying; the former from the action of the planets upon Jupiter, the latter from the action of the sun and the satellites upon the redundant matter accumulated round his equator. Laplace took into account the effects of both these changes in computing the latitudes of the satellites.

An interesting account of the perturbations of Jupiter's satellites will be found in Airy's Treatise on Gravitation.

+ Méc. Cel, tome iv., Pref. p. xiv.

ticity. The fourth datum was the equation of the centre of the third satellite, depending on the position of the perijove of the fourth. The fifth was the motion of the apsides of the fourth satellite.

By comparing these data with his analytical formulæ, Laplace determined the masses of the satellites and the ratio between the equatorial and polar axes of their primary. It appeared from his results that the third satellite contained the greatest quantity of matter, and the first satellite the least *. The mass of the third satellite was found to be about double the moon's mass, and that of the fourth satellite was equal to it. By assuming the equatorial axis of Jupiter to be equal to unity, he found that the polar axis was equal to .9286. It hence followed that the lengths of the polar and equatorial axes were very nearly as 13 to 14; a result which almost coincides with that derived from micrometric measurements of the two axes. Laplace, indeed, considers that in this case theory conducts to a more accurate result than direct observation. It is assuredly one of the greatest triumphs that the human mind can boast of, to have been enabled to determine the precise shape of the planet, by merely observing the eclipses of the small bodies which circulate round him.

Delambre determined the thirty-one elements of Laplace's theory by comparing together a vast number of eclipses observed by astronomers at different periods. Having executed this important task, he then computed the numerical values of all the equations, and employed them in the construction of ecliptic tables of the satellites. These tables were inserted in the third edition of Lalande's Astronomy, which was published in the year 1792, and were found to surpass greatly in accuracy the tables of Wargentin, and those of all preceding astronomers.

The eclipses of the first satellite originally led to the discovery of the successive propagation of light, and this important doctrine was afterwards established upon an indisputable basis by Bradley's discovery of aberration. Laplace, however, conceived that the order of the inquiry might be inverted, by deriving the maximum value of aberration from the velocity of light, as indicated by the eclipses of the satellite. Having suggested this view of the question to Delambre, that indefatigable astronomer undertook the laborious task of computing the velocity of light by the discussion of a great number of eclipses; and from the result obtained by him, he concluded that the maximum value of aberration is equal to 20".25. This value agrees precisely with that which Bradley derived from direct observations on a great number of stars. It is interesting to trace so close an agreement between two methods so widely different. This coincidence shews that the motion of light is uniform within the earth's orbit, for the aberration is derived, in the one case from the velocity of light in the earth's orbit, and, in the other case, from the time which it takes in traversing the diameter of the orbit. Its motion is also uniform within the orbit of Jupiter, for the variations of the radius vector of the planet are very sensible, and the differences in the times of eclipses which these occasion are found to correspond exactly with the supposition of the uniform motion of light.

The following are the values of the masses of the satellites as given by Laplace in the Mécanique Céleste, liv. viii. chap. viii., Jupiter's mass being supposed equal to unity.

CHAPTER IX.

Secular Variations of the Planets.-Elements of the Terrestrial Orbit. Variations of the Eccentricity.-Motion of the Aphelion.-Obliquity of the Ecliptic.--Its secular Variation computed by Theory.-Euler.-Lagrange.-Laplace. - Influence of the displacement of the Ecliptic on the length of the tropical Year.-Indirect Action of the Planets on the terrestrial Spheroid.—Its effect in restricting the Variations of the Obliquity of the Ecliptic and the length of the tropical Year.-Invariable Plane of the Planetary System.-Theory of Comets.-Hevelius.-Borelli.-Dörfel.-Subjection of the Motions of Comets to the theory of Gravitation by Newton.— Halley.—Clairaut.—Researches of Lagrange on Cometary Perturbation.-Lexell's Comet.-Its Perturbations investigated by Laplace.-Publication of the Mécanique Céleste.-General Reflections on the Progress of Physical Astronomy.

THE theory of the secular variations of the elements of the planetary orbits forms one of the most interesting subjects of physical astronomy. The actual existence of some of these variations was long a disputed point with astronomers; but they have been established beyond all doubt in recent times by the accuracy of modern observations. The secular variations of the terrestrial orbit have naturally excited a more lively interest than the others of the same class, on account of their connexion with the physical condition of the earth. The investigation relative to the eccentricity is manifestly an object of the highest importance, since the indefinite increase of that element, at however slow a rate, would ultimately occasion such violent alternations of heat and cold at the earth's surface, in the course of every year, as utterly to destroy the existing economy of animal and vegetable life. The sublime researches of Lagrange have shewn, however, that such a condition cannot possibly ensue; for the terrestrial eccentricity will always be maintained by the action of the planets within certain narrow limits between which it will perpetually oscillate. At present it is diminishing at the rate of 18" in a century. An immense number of ages will elapse before it reach its minimum state; but, when this takes place, it will then pursue a contrary order of variation, increasing at the same slow rate as that at which it had previously diminished. We have seen that the variation of this element forms the medium through which the action of the planets is propagated to the moon; occasioning thereby the secular inequality in the mean motion of that body, which was so long the cause of embarrassment to mathematicians and astronomers, until its physical origin was at length discovered by Laplace.

The motion of the earth's aphelion was first discovered by the celebrated Arabian astronomer, Al Batani. As in the case of the lunar apogee, it advances in the order of the signs, though at a much slower rate. Its annual motion is estimated at 11" or 12". The variation of this element is interesting on account of the clear evidence it affords of the disturbing action of the planets on the earth; but it obviously cannot exercise any influence on the physical condition of the latter.

But the question is very different when we consider the position of the earth's orbit. A variation of this kind, by altering the obliquity of the ecliptic, would manifestly affect the temperature and climate of the earth in an equal degree with the variation of the eccentricity. The true state of the obliquity of the ecliptic was long a subject of controversy; some astronomers asserting that it was invariable, while others maintained that

H

it was constantly diminishing. The earliest measurement of it, if we exclude the records of Eastern nations, is due to Eratosthenes, who flourished about the year 270 a.c. This astronomer found the angle between the tropics to amount to 47° 42′ 27′′, whence the obliquity of the ecliptic was equal to 23° 51′ 13′′. Al Batani, in the ninth century, fixed it at 23° 35'. Waltherus, the German astronomer, made it 23° 29′ 47′′ about the close of the fifteenth century; and Tycho Brahé made it 23° 29′ about the year 1581. Riccioli, Gassendi, and Flamstead maintained that the obliquity was invariable, ascribing the discordances of astronomers wholly to errors of observation. On the other hand, Bouillaud and Wendelin contended that it was continually diminishing; and this opinion was urged with great ability by Louville, in the Memoirs of the Academy of Sciences for 1716. About the middle of the last century, Bradley, Lacaille, and Mayer found the obliquity to be 23° 28′ 18′′, and in 1800, Delambre, Mechain, and Lefrançius made it 23° 28'. Thus it appears that, although the earlier observations are not entitled to much confidence when considered by themselves, the aggregate of the results indicates beyond all question a constant diminution of the obliquity.

Euler first explained the variation of the obliquity of the ecliptic by the theory of gravitation. In his memoir of 1748, he showed that the action of Jupiter on the earth would occasion a displacement in the plane of the ecliptic; tending to bring it nearer to the equator. He investigated the same subject more completely in the Berlin Memoirs for 1754, and also in his memoir on the perturbations of the Planets, which was crowned by the Academy of Sciences of Paris in 1756. On the last-mentioned occasion he made the secular diminution equal to 48", a quantity which differs only about 2" from the most recent determinations of astronomers. Lagrange computed the diminution of the obliquity in the Berlin Memoirs for 1782, and obtained 61".5 for the amount of the secular variation. This result is universally allowed to be too great. The source of Lagrange's error doubtless lay in the erroneous value which he assumed for the mass of Venus, the planet which exercises the greatest influence on the position of the ecliptic.

An interesting question arises; will the obliquity continually diminish until the equator and ecliptic coincide? If this should happen, the sun will daily attain the same meridional altitude as at the equinoxes, and an eternal spring will reign over the whole earth. Lagrange first shewed that such a condition cannot possibly exist; the mutual action of the planets occasioning only small oscillations in the positions of their orbits. The ecliptic will, therefore, continue to approach the equator until it reach the limit assigned by the action of the perturbing forces, after which it will gradually recede from that plane according to the same law as that which determined its previous approach. The diminution of the obliquity is not uniform; but the law of variation can only be ascertained by theory. The formula for computing the obliquity corresponding to any assigned time may be thus expressed :-8=23° 27′ 54′′.8-0". 488566t0′′. 000005ť", t denoting the number of years before or after 1800. This formula will be accurate enough for all the purposes of astronomy, when the value of t does not exceed ten or twelve centuries; it will even serve for all the ancient observations, when we take into account the uncertainty that hangs over them. In the lapse of ages the law of variation will be

Bessel in his Tabulæ Regiomontanæ, 1830, makes it 45".7.

more completely developed; and it will be necessary to include in the formula the cubes and higher powers of t, if the same epoch should be always retained.

Laplace compared the preceding formula with an ancient observation recorded in the annals of the Chinese. It appears that Tcheou Kong, the regent of China, measured the summer solstice about the year 1100, before the Christian era, and from the result obtained by him, combined with another recorded measurement of the winter solstice, astronomers have deduced 23° 54′ 2′′.5 as the obliquity of the ecliptic in his time. Now, if we substitute 2900 for t in the preceding formula, applying to it the negative sign, because it represents the number of years by which the recorded observation has preceded the epoch of the formula, we get 23° 51′ 30′′ for the value of the obliquity. The near agreement of this result with that derived from the Chinese records is very remarkable, especially when we consider that the instrument with which Tcheou Kong made his observation was a vertical gnomon only eight feet high. The accordance between theory and observation is still more striking when we institute a comparison between the results of the former and the accurate determinations of modern astronomers. Thus Arago and Mathieu, from observations on the solstices made in the years 1812, 1813, 1814, coneluded that the obliquity for the year 1813 was 23° 27′ 49′′.28. If we compute its value for the same epoch by the above formula, we obtain 23° 27′ 48.69 as our result. The difference between the observed and computed values is, therefore, only 0.59.

The displacement of the ecliptic, caused by the action of the planets on the earth, affects the precession of the equinoxes, and the length of the tropical year. This displacement is found to produce a very slow motion of the equinoctial points upon the plane of the ecliptic in the direction of the earth's motion. Its effect is, therefore, contrary to that produced by the action of the sun and moon upon the terrestrial spheroid; but, as it is very minute compared with the latter, it merely occasions a small diminution of the annual quantity of precession. The mean value of precession, as determined by observation for the epoch of 1800, is 50".22350. The lunisolar precession computed for the same epoch by the theory of gravitation is 50.37315; hence the effect due to the displacement of the ecliptic amounts to 0.14965.

The progression of the equinoxes, caused by the action of the planets on the earth, will manifestly affect the length of the tropical year. The action of the sun and moon upon the terrestrial spheroid accelerates the arrival of the earth in the equinoxes, and, therefore, shortens the year; the action of the planets on the other hand retards its arrival, and consequently lengthens the year. The acceleration being, however, greater than the retardation, the tropical year will, on the whole, be shortened by the motion of the equinoctial points. If this motion was uniform, the length of the tropical year would be invariable; but such, in fact, is not the case. At present it is slowly increasing, and on this account the tropical year is gradually becoming shorter. The rate of diminution is about half a second in a century; and consequently the tropical year is now shorter by about ten seconds than it was in the time of Hipparchus. If the position of the equator was fixed, the variation of the obliquity of the ecliptic would depend entirely on the displacement of the latter plane, occasioned by the action of the planets on the earth. Laplace, however, has shewn that this displacement gives rise to a corresponding