fortunately, the planetary system is so constituted as to favour the researches of the mathematician. The problem of a planet's motion, when considered in its most general sense, requires that we should include in one common investigation the attractive forces exerted upon the planet by the various bodies composing the solar system. The sun, however, exercises such a preponderating influence, on account of his enormous mass, that we may regard each of the planets as revolving round him in an orbit, approaching very closely to an ellipse; while the other planets may be considered as so many perturbing bodies, producing continual irregularities in the elliptic motion. These perturbations being very minute, the action of each planet may be investigated in succession, without taking into account the simultaneous action of the others; and the aggregate of the results so obtained, when applied to the elliptic motion, will determine the true place of the planet in its orbit. The whole question is, therefore, reduced to the investigation of the motion of one body revolving round another, and continually disturbed by the attraction of a third body. Thus originated the famous Problem of Three Bodies, which has formed the basis of so much profound research in physical astronomy. The rigorous solution of this problem has been found to surpass the powers of the understanding, notwithstanding the many improvements which have been effected in the infinitesimal analysis; but the same considerations, which limit the investigation to the mutual attraction of three bodies, conduct also to other important simplifications. The masses of the planets being in fact very small, compared with the sun's mass, and the eccentricities as well as the inclinations of the planetary orbits being also very inconsiderable, a number of terms involving these elements in the general solution of the problem become, in consequence, so small as to admit of being rejected; and the geometer is thereby enabled to bring the subject within the reach of his analysis. Notwithstanding these obvious advantages, the utmost resources of a profound calculus, combined with the most consummate analytical skill, are indispensably required, in order to effect a solution of this difficult problem; and even then the object can only be attained by a process of successive approximation. In the lunar theory, the principal disturbing body is the sun; for the planets are either too small or too remote to exercise much influence. It might naturally be supposed that the sun, on account of his enormous mass, would very much derange the moon's motion; but in reality the effect of his attractive power is greatly diminished by the immense distance at which he is placed compared with the earth, which is in this case the central body. Still the inequalities of the moon's motion are much more considerable than the perturbations which take place in the motions of the planets; and, on this account, they were justly considered to afford the most favourable means of testing the theory of gravitation. We have already alluded to the failure which attended Newton's attempt to determine the motion of the lunar apogee. Singular enough, when Clairaut and the other two geometers above mentioned deduced the motion of the apogee from their respective analytical solutions of the problem of the lunar perturbations, they found, like Newton, that the result was equal only to half the observed motion. This anomalous fact excited great surprise in the scientific world, and many persons began to entertain a strong suspicion that the law of gravitation, as announced by Newton, was erroneous. Clairaut, despairing of being able to reconcile the ordinary law with the results of observation, proposed that, instead of representing the force by

a term depending on the inverse square of the distance, it should be expressed by two terms, one composed of the inverse square, and the other of the inverse of the fourth power of the distance. Buffon adduced metaphysical arguments against this law; and the question continued to excite a deep interest among men of science. At length Clairaut discovered that, when the lunar perturbations were rightly computed, according to the Newtonian law, the motion of the apogee, when so computed, was exactly conformable to the observed motion.

He found, in fact, by repeating the approximation and taking into account certain small terms which he had previously neglected, that the value obtained by him in the first instance was now exactly doubled. D'Alembert and Euler, upon a revisal of their labours, arrived at the same conclusion; and thus a circumstance, which at one time threatened to subvert the whole structure of the Newtonian theory, resulted in becoming one of its strongest confirmations.

It is right, also, to mention that Thomas Simpson arrived at the correct motion of the apogee before he learned the successful result of Clairaut's labours. This eminent analyst might have done much to sustain the reputation of his country in the researches of physical astronomy if he had lived under more auspicious circumstances.

The method of lunar distances, which offers such advantages in finding the longitude at sea, rendered an accurate knowledge of the moon's motion peculiarly desirable. In 1754, Clairaut and D'Alembert published lunar tables based upon their respective theories. Those of Clairaut obtained considerable credit for accuracy; but D'Alembert's efforts were less fortunate, chiefly in consequence of having paid too little attention to observation in the evaluation of his coefficients. In 1755 Euler published his researches in the lunar theory, accompanied with new tables, greatly superior in accuracy to those of 1746. In his analysis he resolved the forces acting upon the moon along three rectangular co-ordinates, after the example of Maclaurin, who, a few years previously, had first employed this method in his elegant geometrical investigations connected with the question of the Tides. In 1772 he published a third set of tables, based upon a most elaborate analysis of the moon's motion; but, notwithstanding the amount of thought expended on them, they proved inferior in point of accuracy to those of Mayer, chiefly in consequence of his having placed too much reliance on theory in fixing the maximum values of the equations. Mayer, to whom allusion has been already made, was the first person who constructed lunar tables of sufficient accuracy for the great practical purpose of finding the longitude at sea. This he did in 1755, by means of Euler's theory and a skilful discussion of observations. These tables were found to come within the limit of accuracy fixed by the Board of Longitude of this country; and a recompense of £3000 was in consequence awarded to the widow of Mayer, the astronomer himself having died some years before this decision was come to. Bradley, who was appointed to compare the tables with observation, states, in his report of them, that in no case did the error exceed 1. They were first printed in the year

1770.

CHAPTER IV.

Perturbations of the Planets. Inequality of Long Period in the Mean Motions of Jupiter and Saturn.-Researches of Euler.-Perturbations of the Earth.-Clairaut.-Perturbations of Venus.-Lagrange. His Investigation of the Problem of Three Bodies.- Secular Variations of the Planets.—Laplace.-His Researches on the Theory of Jupiter and Saturn. Invariability of the Mean Distances of the Planets.- Oscillations of the Eccentricities and Inclinations.-Stability of the Planetary System.

THE planets, while revolving round the sun, continually disturb each other by their mutual attraction, and hence arise numerous inequalities in their motions, similar to those which take place in the motion of the moon round the earth. Although these disturbing forces form a class of relations as complicated as the mind can perhaps imagine, the study of their effects is on many accounts peculiarly attractive to the thoughtful enquirer. The fundamental ideas are clear and well defined; the principles are firmly established, the methods of research are derivable wholly from the resources of the intellect, and the subject is both vast in extent and varied in character. The magnificent prizes which the theory of gravitation offers prospectively to the mathematician, as the rewards of his labours, are also calculated to allure his researches, while its extreme intricacy serves only to redouble his energies, and stimulate his inventive powers. Hence Physical Astronomy is characterized by a multitude of conceptions at once ingenious, subtle, and profound, while its investigations are pursued, throughout their long and intricate windings, with a coherence and beauty of ratiocination unequalled in any other branch of Natural Philosophy.

Apart from those more obvious questions which impart an interest to the study of Celestial Mechanics, others of the highest moment, with respect to the stability of the system of the world, are also involved in the subject. These questions naturally offered themselves to mathematicians, while engaged in researches connected with the actual motions of the planets, and continued for some time to form the subject of profound study. Their complete solution will ever be ranked among the most brilliant triumphs recorded in the annals of science, while it has shed an imperishable lustre on the names of those eminent individuals by whose labours it has been achieved.

The masses of the planets being small compared with the mass of the central body, the derangements occasioned by their mutual attraction do not in any case attain a magnitude comparable to that of the lunar inequalities. Indeed their existence has generally been established only by a comparison of distant observations, conducted with all the refinements of practical astronomy. In many instances theory has preceded observation, and has pointed out inequalities which, on account of their extreme minuteness, might otherwise have for ever escaped detection. The planets Jupiter and Saturn, being favourably placed in the system for the exertion of their mutual attraction, and their masses being also considerable, it might be expected that the inequalities of their motions would be more readily appreciable than those of the other planets. In fact, as early as 1625, Kepler remarked that the observed places of these planets could not be reconciled with the usually admitted values of their mean motions.

The errors of both planets were found to increase continually in the same direction, with this difference, that the tables made the mean motion of Jupiter too slow, and that of Saturn too quick. Lemonnier found that, by adopting the mean motion of Saturn, as determined by a comparison of ancient with modern observations, the planet had fallen behind its computed place to the extent of 2' in 1598, 20' in 1657, and 364 in 1716. Halley first suspected that the anomalous irregularities of the two planets were due to their mutual attraction. He also attempted to determine the magnitude of the inequality for each planet. He concluded from his researches that in 2000 years the acceleration of Jupiter amounted to 3° 49′, and the retardation of Saturn to 9° 16'. In his tables of the planets he represented the errors by two secular equations increasing as the square of the time, the one being additive to the mean motion of Jupiter, and the other subtractive from the mean motion of Saturn.

The Academy of Sciences of Paris, desirous of obtaining an explanation of these inequalities, in accordance with the theory of gravitation, offered its prize of 1748 for their complete investigation. Euler was induced to compose a memoir on the subject, which was crowned by the Academy; but, although his researches contain a valuable exposition of the analytical theory of planetary perturbation, he was unable to throw any light on the main object of the inquiry. He found a series of inequalities in the mean motions of both planets, but they were all such as completed the cycles of their values every time that the planets returned to the same configurations. He concluded, therefore, that the observed irregularities must be attributed to some extraneous cause, and not to the mutual attraction of the two planets.

Euler in this memoir resolved the differential equation, relative to the latitude of the disturbed planet, into two differential equations of the first order, one of them expressing the differential of the inclination, and the other that of the planet's distance from the node. This may be considered as the germ of the famous method of the variation of arbitrary constants.

The theory of Jupiter and Saturn offers some difficulties of a peculiar kind, which did not occur in the investigation of the lunar inequalities. The disturbing action of one body upon another may be expressed by a series of terms involving the ratio of the mean distances of both from the central body. In the lunar theory this fraction is very small, on account of the great distance of the sun, which is the disturbing body; hence the terms converge with great rapidity, and an approximate value of the series is readily obtained. When the question, however, refers to the mutual action of Jupiter and Saturn, the same fraction rises to a considerable maguitude, and the terms of the series converge in consequence with such extreme slowness, as to render impracticable the usual method of computation. Euler's genius was eminently conspicuous in devising the means of vanquishing this difficulty, which would effectually have obstructed a mind gifted with less fertile powers of invention.

The explanation of the motion of the lunar apogee by Clairaut in 1749, having inspired renewed confidence in the principle of gravitation, as adequate to account for all the phenomena of the planetary motions, the Academy of Sciences was again induced to propose the theory of Jupiter and Saturn as the subject of their prize for 1752. Euler was on this occasion also the successful competitor, but he now actually discovered secular equations in the mean motions of both planets, depending on the angular distance

between the aphelia of their orbits. Contrary to observation, however, he found that the two equations were equal in magnitude, and were in both cases additive to the mean motion. He fixed the inequality at 2′ 24′′ for the first century, counting from 1700. Nothwithstanding the analytical skill which this geometer displayed in his researches, he signally failed in his efforts to account for the irregularities of the two planets by the Newtonian theory; and their physical origin, therefore, still continued to be involved in profound mystery.

The attention of geometers was now directed to the perturbations in the earth's motion occasioned by the other planets. Euler investigated. this subject in an elaborate memoir, which was crowned by the Academy of Sciences in the year 1756. It was on this occasion that he explained and partially developed the theory of the variation of arbitrary constants. In considering the motion of a planet in an elliptic orbit, there are six constants or elements, which by their independent variations would severally modify the motion. These are-1°, the major axis of the orbit, or the mean distance; 2° the eccentricity; 3°, the position of the line of apsides; 4o, the inclination of the orbit with respect to a fixed plane; 5o, the position of the line of nodes; 6°, the longitude of the planet at any assigned instant, or the longitude of the epoch, as it is called. Now if the planet were exposed only to the action of the sun these elements would remain invariable, and the planet would continually revolve in the same ellipse. Its place, corresponding to any given time, might therefore be readily computed, by means of Kepler's law of the areas, when once these six elements were known. As, however, it is continually disturbed in its motion by the action of the other planets, the theory of a constant ellipse will no longer be applicable to the question. Still, as its aberrations from an elliptic orbit are very small, its place may be computed by assuming it to move in a mean ellipse, and then ascertaining the minute irregularities occasioned by the perturbing forces. This is the course which geometers had hitherto pursued in all researches connected with the problem of three bodies. Euler, however, proposed to compute the motion wholly by the elliptic theory, upon the supposition that the planet continually revolved in an ellipse, the elements of which varied every instant from the action of the other planets. By these means the whole effect of perturbation was thrown upon the elements of the orbit, and when these were ascertained for any given instant, it was easy to calculate the corresponding place of the planet by the elliptic theory alone. As this refined conception has not unfrequently been ascribed to Lagrange, it may be proper to cite Euler's own words in reference to it. After obtaining the differential expressions of the elements, he then proceeds in the following terms to point out their advantages: These formulæ appear to be peculiarly commodious in computing the deviations of the motion from Kepler's laws; since they have reference to motion in an ellipse, which varies continually, as well in respect to the parameter as to the eccentricity and the position of the apsides. For, during an indefinitely small portion of time, the motion of the planet may be conceived as taking place in an ellipse, according to the laws of Kepler; and, if the elements of this ellipse be computed for any given time, by means of the formulæ just found, the true place of the planet, relative to an assumed plane, may be also assigned."* This investigation of Euler's, like the two previous ones, displays abun

66

• Prix de l'Académie, tome viii. Investigatio Motuum Planetarum, p. 29.

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