LAPLACE. (continued 2)

Diminution of the obliquity of the ecliptic.—The astronomers of the eighteenth century had found, by a comparison of ancient with modern observations, that the obliquity of the ecliptic is slowly diminishing from century to century. The researches of geometers on the theory of gravitation had shown that an effect of this kind must be produced by the disturbing action of the planets on the earth. Laplace determined the secular displacement of the plane of the earth's orbit due to each of the planets, and in this way ascertained the whole effect of perturbation upon the obliquity of the ecliptic. A comparison which he instituted between the results of his formula and an ancient observation recorded in the Chinese Annals exhibited a most satisfactory accordance. The observation in question indicated the obliquity of the ecliptic for the year 1100 before the Christian era, to be 23° 54' 2".5. According to the principles of the theory of gravitation, the obliquity for the same epoch would be 23° 51' 30".

Limits of the obliquity of the ecliptic modified by the action of the sun and moon upon the terrestrial spheroid.—The ecliptic will not continue indefinitely to approach the equator. After attaining a certain limit it will then vary in the opposite direction, and the obliquity will continually increase in like manner as it previously diminished. Finally, the inclination of the equator and the ecliptic will attain a certain maximum value, and then the obliquity will again diminish. Thus the angle contained between the two planes will perpetually oscillate within certain limits. The extent of variation is inconsiderable. Laplace found that, in consequence of the spheroidal figure of the earth, it is even less than it would otherwise have been. This will be readily understood, when we state that the disturbing action of the sun and moon upon the terrestrial spheroid produces an oscillation of the earth's axis which occasions a periodic variation of the obliquity of the ecliptic. Now, as the plane of the ecliptic approaches the equator, the mean disturbing action of the sun and moon upon the redundant matter accumulated around the latter will undergo a corresponding variation, and hence will arise an inconceivably slow movement of the plane of the equator, which will necessarily affect the obliquity of the ecliptic. Laplace found that if it were not for this cause, the obliquity of the ecliptic would oscillate to the extent of 4° 53' 33" on each side of a mean value, but that when the movements of both planes are taken into account, the extent of oscillation is reduced to 1° 33' 45".

Variation of the length of the tropical year.—The disturbing action of the sun and moon upon the terrestrial spheroid occasions a continual regression of the equinoctial points, and hence arises the distinction between the sidereal and tropical year. The effect is modified in a small degree by the variation of the plane of the ecliptic, which tends to produce a progression of the equinoxes. If the movement of the equinoctial points arising from these combined causes was uniform, the length of the tropical year would be manifestly invariable. Theory, however, indicates that for ages past the rate of regression has been slowly increasing, and, consequently, the length of the tropical year has been gradually diminishing. The rate of diminution is exceedingly small. Laplace found that it amounts to somewhat less than half a second in a century. Consequently, the length of the tropical year is now about ten seconds less than it was in the time of Hipparchus.

Limits of variation of the tropical year modified by the disturbing action of the sun and moon upon the terrestrial spheroid.—The tropical year will not continue indefinitely to diminish in length. When it has once attained a certain minimum value, it will then increase until finally having attained an extreme value in the opposite direction, it will again begin to diminish, and thus it will perpetually oscillate between certain fixed limits. Laplace found that the extent to which the tropical year is liable to vary from this cause, amounts to thirty-eight seconds. If it were not for the effect produced upon the inclination of the equator to the ecliptic by the mean disturbing action of the sun and moon upon the terrestrial spheroid, the extent of variation would amount to 162 seconds.

Motion of the perihelion of the terrestrial orbit.—The major axis of the orbit of each planet is in a state of continual movement from the disturbing action of the other planets. In some cases, it makes the complete tour of the heavens; in others, it merely oscillates around a mean position. In the case of the earth's orbit, the perihelion is slowly advancing in the same direction as that in which all the planets are revolving around the sun. The alteration of its position with respect to the stars amounts to about 11" in a year, but since the equinox is regressing in the opposite direction at the rate of 50" in a year, the whole annual variation of the longitude of the terrestrial perihelion amounts to 61". Laplace has considered two remarkable epochs in connection with this fact; viz: the epoch at which the major axis of the earth's orbit coincided with the line of the equinoxes, and the epoch at which it stood perpendicular to that line. By calculation, he found the former of these epochs to be referable to the year 4107, b.c., and the latter to the year 1245, a.d. He accordingly suggested that the latter should be used as a universal epoch for the regulation of chronological occurrences.

(B.)

The Mécanique Céleste.—This stupendous monument of intellectual research consists, as stated by the author, of five quarto volumes. The subject-matter is divided into sixteen books, and each book again is subdivided into several chapters. Vol. I. contains the first and second books of the work; Vol. II. contains the third, fourth, and fifth books; Vol. III. contains the sixth and seventh books; Vol. IV. contains the eighth, ninth, and tenth books; and, finally, Vol. V. contains the remaining six books. In the first book the author treats of the general laws of equilibrium and motion. In the second book he treats of the law of gravitation, and the movements of the centres of gravity of the celestial bodies. In the third book he investigates the subject of the figures of the celestial bodies. In the fourth book he considers the oscillations of the ocean and the atmosphere, arising from the disturbing action of the celestial bodies. The fifth book is devoted to the investigation of the movements of the celestial bodies around their centres of gravity. In this book the author gives a solution of the great problems of the precession of the equinoxes and the libration of the moon, and determines the conditions upon which the stability of Saturn's ring depends. The sixth book is devoted to the theory of the planetary movements; the seventh, to the lunar theory; the eighth, to the theory of the satellites of Jupiter, Saturn, and Uranus; and the ninth, to the theory of comets. In the tenth book the author investigates various subjects relating to the system of the universe. Among these may be mentioned the theory of astronomical refractions; the determination of heights by the barometer; the investigation of the effects produced on the movements of the planets and comets by a resisting medium; and the determination of the values of the masses of the planets and satellites. In the six books forming the fifth volume of the work, the author, besides presenting his readers with an historical exposition of the labours of Newton and his successors on the theory of gravitation, gives an account of various researches relative to the system of the universe, which had occupied his attention subsequently to the publication of the previous volumes. In the eleventh book he considers the subjects of the figure and rotation of the earth. In the twelfth book he investigates the attraction and repulsion of spheres, and the laws of equilibrium and motion of elastic fluids. The thirteenth book is devoted to researches on the oscillations of the fluids which cover the surfaces of the planets; the fourteenth, to the subject of the movements of the celestial bodies around their centres of gravity; the fifteenth, to the movements of the planets and comets; and the sixteenth, to the movements of the satellites. The author published a supplement to the third volume, containing the results of certain researches on the planetary theory, and a supplement to the tenth book, in which he investigates very fully the theory of capillary attraction. There was also published a posthumous supplement to the fifth volume, the manuscript of which was found among his papers after his death.