perturbations of the moon's motions. At nearly the same time two of the ablest mathematicians of France, Clairault and D'Alembert, also undertook the same problems without any knowledge of each other's intentions.*

mers.

By comparing together the observations which had been made since the re-establishment of astronomy, it was found that the motion of Jupiter was more rapid, while Saturn moved more slowly, than was indicated by comparing the same observations with those made by the ancient astronoThese motions of the two great planets offered to the mathematician a problem worthy of his best efforts. The Academy of Sciences of Paris, accordingly, offered their prize for 1748 for an investigation of the cause of these motions. If there were but two bodies, as, for instance, the sun and the earth, it had been shown by Newton that, for this case, both bodies would describe elliptical orbits about their common centre of gravity, which point would occupy a common focus of the two ellipsis. Since the sun's mass is immensely greater than the mass of any of the planets, the motion is principally thrown upon the smaller body, that of the sun being scarcely perceptible. If we now introduce a third body and its influence into our calculation, the problem of the motion of any one of them becomes very complicated; and if the masses of the several bodies were nearly equal to one another, its solution would surpass the present powers of analysis. But in the planetary system the mass of the sun is so great as compared with the mass of any one or of all the planets, that the influence which our planet can have on the motions of another can only produce minute irregularities, which it thus becomes possible to calculate by methods of approximation which are employed. If the third body attracted the other two with the same force and in the same direction, it would not disturb their relative motions; but since that is not the case, it is the difference of the forces, estimated in the same direction, which act on the two bodies, that produce irregularities or perturbations. It is this difference that is called the disturbing force.

*Grant's Hist. Phys. Ast., p. 44.

It is easy to find the value of the disturbing force, but when thus generally found, it is not in a condition to be employed in the actual calculation of the motions of a planet, and it becomes necessary to resolve it into a series of other forces, which decrease in value, and which can be employed. These partial forces, into which the complete value of the disturbing force is resolved, are expressed either by the sine or the cosine of an arc that increases uniformly with the time. The original series into which the disturbing force is resolved, is usually arranged according to the ascending powers of the ratio of the distances of the disturbing and of the disturbed planet from the sun, the ratio being in every case much less than unity.

In the case of the moon disturbed by the sun, the ratio to which we have referred is very small, being only about onefour-hundredth, the distance of the sun from the earth being about four hundred times as great as the distance of the moon from the earth. A very few terms of the series will suffice in the lunar theory; but in the planetary theory the ratio in question is in some cases (Venus and the earth, for instance) nearly equal to three-fourths, and a dozen or more terms have to be employed. Owing to this circumstance, the planetary theory, from the great number of terms of the completely developed series, becomes very complicated.

It was natural to conclude that the cause of the irregularities in the motions of Jupiter and Saturn was owing to their mutual attraction. Euler, Clairault, and D'Alembert attempted the solution of the problem upon this principle. Clairault first undertook to calculate the irregularities after the manner of Newton, but he met with insuperable difficulties, and finally abandoned that method for the analytical. Each of the great geometers to whom we have referred produced a memoir in which the perturbations of the motions of the planets Jupiter and Saturn were investigated by means of analytical processes. Euler's memoir was crowned by the Academy. His researches contain a valuable exposition of the theory of planetary perturbation, but he failed to throw any light on the main object of inquiry. His profound skill in analysis enabled him to

surmount difficulties that would have arrested the progress of all ordinary geometers. This memoir of Euler contains the first germ of the very important method of the variation of arbitrary constants, a method which we must now explain.

Since the ellipse approximates the most nearly of any known curve to the orbit actually described by any one of the known planets, we may imagine an ellipse described so as to pass through any two consecutive points of the orbit. Since the disturbing force is very small when compared with the central force of the sun, which retains the planet in its orbit, the imaginary ellipse, which we have supposed drawn, cannot differ very much from the ellipse which would be described if the disturbing force did not exist. If we, therefore, suppose this ellipse to expand or contract, and to change its form, according to circumstances, so as to adapt it to the actual place of the planet, it is evident that the planet will be found at the same time in the orbit which it actually describes, and in this variable imaginary ellipse. The difliculty will now consist in calculating the magnitude, form, and position of the ellipse. This we may do by considering the elements of the planet's orbit-the major axis, the eccentricity, the longitude of the perihelion, the inclination of the plane of the orbit to a given fixed plane, and the longitude of the ascending node-for a known epoch, as variable. If it were possible to calculate the amount of the variation of each element, this correction being applied to the known elements would make known the elements of our imaginary ellipse, and it would thence be possible to calculate the actual position of the planet in its orbit. The method of the variation of arbitrary constants consists in find. ing analytical expressions for the variation of the known elements of a planet's orbit. The constitution of the solar system is such that, in most cases, we can estimate and apply the effects of each disturbing cause separately.

Nature actually suggests the method of variation of arbitrary constants, since, if the elements of a planet's orbit be compared with one another at different epochs considerably separated in time, they are found to have varied. We have mentioned Euler as the first that employed the method of

variation of constants; but we must not omit to mention that Newton actually anticipated all others, in a measure, by representing the paths of bodies by means of revolving orbits.* Still, Euler was the first to treat the subject distinctly as the variation of arbitrary constants. This he did in his memoir on the perturbations of the earth's motion, crowned by the Academy of Sciences, in the year 1756.

The perturbations of the motions of the planets are naturally divided into two general classes, periodical and secular. The periodical inequalities are dependent on the relative positions of the planets, and they pass through their periods in a comparatively short space of time, or when the planets return to their same relative positions. The secular inequalities are independent of the mutual positions of the planets, but depend on the relative sitnation of the orbits of the planets. Or, in other words, the secular inequalities result from the small uncompensated or outstanding quantities, which are left when the planets return to their same relative situations. These uncompensated quantities show themselves by affecting the elements of the orbits of the planets. This class of inequalities is by far the most important when all time is considered, since their continual accumulations in the same direction would ultimately destroy the dynamical stability of our system. It therefore becomes an interesting question to the physical astronomer to ascertain whether the secular inequalities will always continue in the same direction, or whether they are also periodical, and will only cause the elements of the planets to oscillate between fixed limits of mode

rate extent.

Euler sought among the secular terms for an explanation of the long inequality in the mean motions of Jupiter and Saturn, to which we have referred; but, as also stated, he was not successful. In his memoir of 1752, on Jupiter and Saturn, he found the mean motion to be affected by secular inequali

* Principia, Bk. I., Sec. ix.

Laplace says, after giving an abstract of Euler's Memoir of 1749, "C'est le premier essai de la method de la variation des constantes arbitraires." Mec. Cel., livre XV., tome v., p. 305.

ties; but subsequent investigations have shown that Euler, in this case, arrived at erroneous results. In the year 1763, Lagrange, then twenty-seven years of age, gave, in the volume of the Turin Memoirs for that year, a new solution of the problem of three bodies, which he applied to Jupiter and Saturn. He found two secular inequalities, one addition to the mean motion of Jupiter, and the other about five times as great, and subtractive from the mean motion of Saturn. This agreed better with observation than Euler's result, which made both equations equal and addition. Lagrange did not succeed, however, in rendering a complete account of the irregularities in the mean motions of these two bodies.

In the year 1773,* Laplace, then twenty-four years of age, appeared as the rival of Lagrange. He discovered that, by neglecting terms of the fourth order of small quantities, the mean motions of any two planets of our system are not affected by secular inequalities. In the year 1776 Lagrange again applied himself to the investigation of the secular inequalities, by treating the subject by the method of the variation of arbitrary constants. By a comparatively simple analysis he found that the mean distances of the planets, and consequently their mean motions (since the mean motion and time of revolution depend wholly on the mean distances, or the semi-major axes of their orbits), are not subject to any secular inequalities whatever, but are influenced only by periodical inequalities, which depend, as we have stated, on the mutual configuration of the planets, which, in general, run through their periods in comparatively short spaces of time. The importance of this result is such that it is worth while to dwell on it for a moment. Amid all the changes to which the other elements of the planetary orbits are subjectthe secular changes of the eccentricities, running through hundreds of thousands of years; the secular motions of the perihelia, which in some cases cause them to perform complete revolutions in the heavens; the similar motions of the nodes, and the slow secular changes in the position of the

*Hist. Ind. Sciences, Vol. II., p. 107.