by him to the Academy of Sciences of Berlin*. He derived the analytical expressions of these inequalities from the periodic variations of the elements, and then computed their numerical values for each planet.

The interesting results obtained by Lagrange relative to the stability of the system were founded upon a knowledge of the masses of the several planets. The computation of the masses of those planets that are accompanied by satellites is not a difficult problem, but it is quite different when the question refers to the other planets of the system. Theoretically speaking, the masses of all the planets may be ascertained by observing the effects of their mutual perturbations, but these effects are generally so very minute that they are almost entirely lost in the errors of observation. Lagrange determined the masses of the planets that have no satellites by combining their volumes with their densities, assuming the latter to vary in the inverse ratio of the planet's distance from the sun. This principle, although naturally suggested by the relative densities of the Earth, Jupiter, and Saturn, was, notwithstanding, gratuitously assumed, and therefore the consequences derived from it could not be altogether free of uncertainty. Lagrange, indeed, shewed the improbability of any minute alteration in the values of the masses affecting essentially the conclusions at which he arrived; but still it was desirable that such valuable truths should be established by an analysis divested of all considerations of a hypothetic character. This important step was made by Laplacet. In 1784 he demonstrated that, no matter what might be the relative masses of the planets, the eccentricities and inclinations if once inconsiderable would always continue so, provided the planets were subject to this one condition-that they all revolved round the sun in the same direction. This remarkable truth is embodied in two elegant theorems, which the great geometer just mentioned was the first to announce to the world. The theorem relative to the oscillations in the form of the orbits may be thus stated: If the mass of each planet be multiplied by the square of the eccentricity, and this product by the square root of the mean distance, the sum of these quantities will always retain the same magnitude. Now when this sum is determined for any given epoch, it is found to be small; by the preceding theorem, then, it will always continue so; it follows, therefore, à fortiori, that each quantity will continue small, and, consequently, the eccentricity cannot in any case become considerable. The theorem relative to the positions of the orbits is equally elegant. It may be expressed in the following terms: If the mass of each planet be multiplied by the square of the tangent of the orbit's inclination to a fixed plane, and this product by the square root of the mean distance, the sum of such quantities will continue invariable. Considerations similar to those we employed in the previous instance enable us to conclude from this theorem that the orbits of the planets will suffer only a very inconsiderable displacement from their mutual attraction.

• Mém. Acad. Berlin, 1783-4.

† Mém. Acad. des Sciences, 1784. This memoir of Laplace's is remarkable for containing the first announcement of three of the most important discoveries in Physical Astronomy. These were-1st, the explanation of the long inequality of Jupiter and Saturn; 2nd, the investigation of the origin of the curious relations which connect the epochs and mean motions of the three interior satellites of Jupiter; 3rd, the results mentioned in the text.

The value of Laplace's researches on the present occasion does not rest merely on the discovery of the two theorems announced in the text. The fact is, that the investigation of the ultimate condition of the eccentricities and the inclinations depends in each case on

The laws which thus regulate the eccentricities and inclinations of the planetary orbits, combined with the invariability of the mean distances, secure the permanence of the solar system throughout an indefinite lapse of ages, and offer to us an impressive indication of the Supreme Intelligence which presides over nature, and perpetuates her beneficent arrangements. When contemplated merely as speculative truths, they are unquestionably the most important which the transcendental analysis has disclosed to the researches of the geometer, and their complete establishment would suffice to immortalize the names of Lagrange and Laplace, even although these great geniuses possessed no other claims to the recollection of posterity.

It cannot fail to have occurred to the reader that in these sublime researches the two mighty rivals pressed forward always at an equal pace, insomuch that it would be hardly possible for the most discerning judgment to assign the palm of superiority to either of them. Their investigations of the secular variations were in both cases equally original, and equally entitled to admiration. Laplace's method might be more simple; Lagrange's was more luminous, and had the advantage of being direct. his researches connected with the mean motion, Laplace displayed a practical sagacity which rarely characterized the speculations of Euler or Lagrange, and, perhaps, this quality was more valuable to him throughout his career than an unexampled command of analysis was to his great rival. In the integration of the differential equations relative to the secular variations of the planets, the genius of Lagrange was eminently conspicuous. Laplace admits that he was compelled to abandon the design of integrating his own equations on account of the difficulties they offered, and was only induced to resume the subject on becoming acquainted with the ingenious method devised for that purpose by his illustrious contemporary *.

the resolution of an algebraic equation, equal in degree to the number of planets whose mutual action is considered, and involving their masses in indeterminate forms. Lagrange shewed that if any of the roots of this equation should be equal or imaginary, the corresponding element (whether the eccentricity or the inclination) would increase to an indefinite extent, but if the roots should be all real and unequal, the same elements would perpetually oscillate between fixed limits. Having ascertained the masses of the planets by the methods mentioned in the text, he substituted them in each equation, and then, by the method of successive approximation, he obtained the values of the several roots. These he found to be all real and unequal, whether the equation referred to the eccentricities or the inclinations, whence he concluded that these elements would perpetually oscillate. The peculiar merit of Laplace's researches consisted in shewing that the roots were all real and unequal, without having recourse to the actual solution of the equations, and, consequently, without the necessity of employing any determinate values of the masses.

* "Je m'etais proposé depuis long temps de les integrer mais le peu d'utilité de ce calcul pour les besoins de l'Astronomie joints aux difficultés qu'il présentait m'avait fait abandonner cette idée et j'avoue que je ne l'aurais pas reprise sans la lecture d'un excellent mémoire, sur les inegalités seculaires du mouvement des nœuds et de l'inclinaison des orbites des Planetes que M. De Lagrange vient d'envoyer à l'Académie." Mém. Acad. des Sciences, Année 1772, part i. p. 371.

CHAPTER V.

Irregularities of Jupiter and Saturn.-Researches of Lambert.-Lagrange.-Circumstances which determine the Secular Inequalities in the Mean Longitude.-Laplace's Investigation of the Theory of Jupiter and Saturn.-His Discovery of the physical cause of the Long Inequality in their Mean Motions.-Acceleration of the Moon's Mean Motion.-Halley.-Dunthorne.-Failure of Euler and Lagrange to account for the Phenomenon. - Its explanation by Laplace. - Secular Inequalities in the Moon's Perigee and Nodes.-Inequalities depending on the Spheriodal Figure of the Earth.Parallactic Inequality.

ALTHOUGH the principle of gravitation was shewn to be admirably calculated for maintaining the stability of the solar system, the strange irregularities in the mean motions of Jupiter and Saturn still continued to perplex astronomers, and in some degree to tarnish the lustre of the Newtonian theory. In 1773 Lambert published an interesting essay on this subject, in which he attempted to represent the inequalities of the planets by means of empiric equations. The researches of this astronomer contributed to throw some light upon the real character of the phenomenon. It had been hitherto supposed that the mean motion of Jupiter was continually accelerated, and that of Saturn similarly retarded. These results were derived from a comparison of the observations cited by Ptolemy in the Syntaxis, and those of the earlier astronomers of Europe, with the observations of modern times. Lambert, however, found, on comparing the observations of Hevelius with those of the following century, that the mean motion of Jupiter was retarded, while that of Saturn was accelerated. This important fact indicated that the inequalities did not increase indefinitely in the same direction, but were merely periodic, like those depending on the configurations of the planets. The researches of Lagrange, in 1776, tended to strengthen this conclusion; but it is important to remark, that the result he obtained relative to the invariability of the mean distance does not necessarily exclude the existence of secular inequalities in the mean motion. When, indeed, we consider a single planet revolving round the sun in an undisturbed orbit, the mean motion will depend solely on the mean distance, and will not in any manner be affected by the elements which prescribe the form and position of the orbit. Thus we may make the eccentricity and the other elements vary in any manner we please, but so long as the mean distance retains the same magnitude the mean motion will continue unalterable. The relation which connects these two elements forms the third of Kepler's famous laws, and when one of them is known the other is readily deducible from it by means of that relation. But the case will be quite different when we suppose the planet to be perpetually disturbed in its orbit by the action of another planet. The two elements will no longer be connected together by Kepler's law, for the perturbing forces will now introduce into the expression of the mean motion a class of terms depending upon the eccentricities and the other elements of both planets. These elements, in virtue of their secular variations, might produce an effect on the planet's longitude which would ultimately become sensible, and hence might arise a secular inequality in the mean motion, notwithstanding the invariability of the mean distance. It became, therefore, an object of

the highest importance to ascertain whether the mean motions of the planets were affected in this manner, and if so, to determine, in the case of Jupiter and Saturn, whether the effects were of such a magnitude as to account for the observed irregularities of the two planets.

In 1783 Lagrange investigated this interesting question. He had previously found that if all the terms exceeding the first powers of the eccentricities and inclinations were neglected, the perturbing forces would not in any manner whatever affect the mean motion. On the present occasion he extended his inquiries to the terms involving the squares of the eccentricities, and he now actually discovered among them a secular equation affecting the mean motion. Applying his formula to the theory of Jupiter and Saturn, he found that the equation was utterly insensible in both planets; for in neither case did it exceed the thousandth part of a second, even when it reached its maximum value *. "This result," says Lagrange, will allow us to dispense with a similar examination of the secular inequalities in the mean motions of the other planets, as we originally proposed to do, for it is easy to predict that the values of the equations will be even less than those we have just found. We may, then, henceforth consider it as a truth rigorously demonstrated, that the mutual attraction of the principal planets cannot produce any sensible alteration in their mean motions."t

This result is one of the most interesting in physical astronomy, and we have seen that the merit of establishing it is almost wholly due to Lagrange. At this stage of the planetary researches it had the effect of narrowing the question relative to the irregularities of Jupiter and Saturn; for it shewed that if these irregularities resulted from the mutual action of the two planets, their explanation must be sought for among the pe riodic terms, and not among those depending on the secular variations of the elements. It is clear, then, that, apart from its intrinsic value, this result must be considered as forming a most important step in the developement of the theory of gravitation.

It appears from Lagrange's words, as quoted above, that he did not consider himself warranted in concluding from his researches that the mean motions of the secondary planets might not be affected with secular inequalities of sensible magnitudes. Unfortunately for his fame, it did not occur to him to apply his formula to the moon, although the secular inequality which astronomers had actually detected in the mean motion of that satellite might have suggested such a step to a mind of much less sagacity than his. By this inadvertence he missed one of the noblest discoveries in physical astronomy, and it happened to him, as on several other occasions, that, while he allowed his brilliant researches to remain comparatively fruitless in his hands, he had the mortification of seeing the prize carried off by his more persevering and ambitious rival.

Geometers being now assured that the mutual attraction of Jupiter and Saturn could not produce an inequality of a secular character in their

The mean longitude of a planet depends upon two elements:-1st, the mean mo◄ tion; 2nd, the mean longitude corresponding to any given epoch, or, more simply, the longitude of the epoch. As the mean motion is supposed to be derivable from the mean distance by Kepler's law, it cannot affect the mean longitude with a secular inequality, in consequence of the invariability of the element upon which it depends. Hence, in the theory of the variation of arbitrary constants, the secular inequality in the planet's motion is ascribed solely to the variation of the longitude of the epoch, the constant forming the sixth element of elliptic motion.

+ Mém. Acad. Berlin, 1783, p. 223.

mean motions, it only remained for them to inquire whether the anomalous irregularities of the two planets might not be explicable by some periodic inequality of long duration. This was the form which the question assumed when Laplace applied the energies of his powerful mind to a rigorous examination of all the circumstances calculated to affect it. He first proceeded to inquire whether the inequalities were connected together by relations similar to those which would ensue on the supposition that they were produced by the mutual action of the two planets. By a very simple analysis he found that the mean motion of Jupiter would be accelerated, while that of Saturn was retarded, and vice versa. He also discovered that, if we only regard inequalities the periods of which are very long, the corresponding derangements of the two planets would always be to each other as the products formed by multiplying the mass of each planet into the square root of its mean distance. He hence easily concluded that the derangement of Jupiter at any time would be to the simultaneous derangement of Saturn very nearly as 3 to 7. Now, by assuming, according to Halley, that the retardation of Saturn in 2000 years amounted to 9° 16', this relation gave him 3° 58′ for the corresponding acceleration of Jupiter, a quantity which differs only by 9/ from the result obtained by Halley.

Being thus furnished with a strong indication that the irregularities of the two planets were due to their mutual attraction, he entered upon a searching inquiry into their real source. This he finally discovered in the near commensurability of the mean motions of the two planets. Five times the mean motion of Saturn is very nearly equal to twice the mean motion of Jupiter. In fact, if n, n' represent the mean motions of the two planets, 5n-2n' is equal only to about 4th of the mean motion of Jupiter. Now, Laplace found that certain terms, involving this quantity in the differential equations of the longitude, would receive, by double integration, the square of the same quantity as divisors, and in consequence would rise very much in value. Terms of this class are, indeed, generally very minute, being only of the order of the cubes of the eccen tricities and inclinations; but Laplace, with characteristic sagacity, suspected that the small divisors they acquired might render them sensible, and that they might possibly explain the irregularities of the two planets. The result of actual calculation entirely confirmed his suspicion. He found that the terms assigned to Saturn an inequality equal to 48' 44", and to Jupiter a contrary inequality equal to 20' 49". The periods of the two inequalities were equal, and amounted to 929 years. They reached their maximum values in the year 1560. The apparent mean motions of the two planets henceforth continually approximated towards their true mean motions, and finally coincided with them in the year 1790. This is the reason why Halley, on comparing ancient with modern observations, found the mean motion of Jupiter to be quicker, and that of Saturn slower, while Lambert, on the other hand, from a comparison of modern observations with each other, arrived at a diametrically opposite conclu

sion.

Laplace found that his equations accounted in a most satisfactory manner for the irregularities of the two planets. Among forty-three oppositions of Saturn which he compared with theory, the error in no case exceeded 2', and generally it fell very far short of that quantity. At a subsequent period of his researches he diminished the errors of both planets to 12", although only a small number of years before the errors in the