plane of the orbits, extending through vast periods of time like the changes in the eccentricities-there are no changes in the mean distances and the mean motions, but those of a periodical character which pass through their periods in a small fraction of the time required for the secular changes. This is of the highest importance to the inhabitants of our planet, since it establishes the invariability of the solar year as regards long periods of time. This results, not from any special provision for man's benefit, but from the immutable laws of nature, of which the law of universal gravitation, discovered by Newton, is the most general.

The long inequality of Jupiter and Saturn being thus excluded from the secular variations of the elements of the planets, it was evident that if it were a result of the mutual attraction of the planets, it was to be sought for among the periodical inequalities. The problem had assumed this form when Laplace applied his great intellect to an examination of the subject. A little consideration on the relative magnitude and direction of the inequalities,* satisfied him that the motions under consideration were not produced by a foreign cause, but their origin must be looked for among the periodical inequalities. By comparing the mean motions of Jupiter and Saturn, he found that five times the mean motion of Saturn is very nearly equal to twice that of Jupiter. He suspected that the terms in the development of the disturbing force which had this difference for an argument (or variable part) would explain the irregularities under consideration. "The probability of this cause, and the importance of the object, determined me," says Laplace,t" to undertake the laborious calculation necessary to determine this question." When the terms were calculated, they were found to be of the order of the cubes of the eccentricities and inclinations, and similar ones exist of the fifth order with respect to the same quantities. These terms, in the differential equations, or the expressions which give the momentary variations of the quantities which

* Système du Monde, Vol. II., p. 50.

+ Sys. du Monde, Vol. II., p. 59.

are being calculated, are generally very minute, since their coëfficients are very small fractions; but when they are integrated, or when the variations which take place in any given space of time, are derived from the momentary variations, these terms acquire small divisors, which increase the numerical value of the coëfficients, and thus bring the terms into importance, and when such terms have to be integrated twice, as is the case in the question under consideration, their value is still further augmented. This investigation enabled Laplace to explain the cause of the long inequality of Jupiter and Saturn, which had given geometers so much trouble. It has a period of about 929 years.*

*

In discussing the theory of any two planets, terms of the character referred to above must be sought for, even if their coefficients are very small in the differential equation. Thus, twice the mean motion of Mercury minus five times the mean motion of Venus is a small quantity, being only about the one-twenty fourth of the mean motion of Mercury; four times the mean motion of the earth differs from the mean motion of Mercury by only a small quantity; twice the mean motion of Mars and once the mean motion of the earth, are nearly the same; three times the mean motion of Mars is nearly the same as once the mean motion of Venus, and twice the mean motion of Neptune is nearly the same as once the mean motion of Uranus. Professor Airy also found that eight times the mean motion of Venus is nearly the same as thirteen times the mean motion of the earth, and this gives rise to an inequality which has about 240 years for its period.

Professor Newcomb, Professor Peirce, and Professor Kirkwood have discovered some equations which exist among the mean motions of the planets Jupiter, Saturn, Uranus and Neptune. Professor Newcomb found that three times the mean motion of Uranus, plus eight times the mean motion of Neptune, equal the mean motion of Jupiter. Professor Kirkwood found that three times the mean motion of Saturn, plus eleven

* See Memoir Acad. Sciences for 1784.

+ Gould's Ast. Jour., Vol. V., p. 101; Silliman's Journal, Jan. 1872, p. 67, and March, 1872, p. 208.

times the mean motion of Neptune, equal twice the mean motion of Jupiter. Several other equations were pointed out. Though these relations will possess but little interest for the general reader, they are very interesting to the physical astronomer, since they will give rise to very important, though in general small terms, when the mutual effects of these planets are simultaneously considered.

In 1782, Lagrange investigated the perturbations of the planets by a method which embraced both the secular and the periodical inequalities in a common analysis. This he did by considering all the perturbations in the motions of the planets as due to the variation of the elements of their orbits. Having found the differential or momentary variations of the elements, he separated them into two 'classes, one dependent on the mutual configuration of the bodies, which gave the periodical variations, and the other dependent on the mutual positions of the orbits, which gave the secular variations.* The results which he obtained coincided with the results which he and Laplace had already found by other methods.

All the general results to which we have now referred were obtained by taking into consideration only the first power of the disturbing forces. That is, when the effect of the variation of the elements, on the succeeding variation, is not taken into consideration, the result is said to be a first approximation, or of the order of the first power of the disturbing force; but when this variation is taken into consideration, it is said to be a second approximation, and the result is of the order of the square of the disturbing force. In the year 1808 M. Poisson, then only twenty-five years of age, attacked the problem of perturbations in this form. His result was, that to the second order of approximation the mean distances are subject to only periodical variations.

This result to which M. Poisson was conducted had the effect to again direct the great masters, Lagrange and Laplace, to the subject of planetary perturbations. The former was then an old man seventy-two years of age, but his mind was

* Mem. Acad. Berlin, 1781, 82, 83.

VOL. XXVI.-NO. LI.

6

yet vigorous, and his great powers of generalization conducted him, as De Morgan says,* to "the most general theorem which has yet been attained in the mathematics of mechanics, not excepting the principle of virtual velocities, or that of D'Alembert." By transferring the origin of coördinates to the centre of gravity of the system of bodies under consideration, his analysis became general, and he showed that even when the variation of the elements of the disturbing planet was taken into consideration, the mean distances were not subject to secular variations. Thus it was demonstrated that to the second power of the disturbing force the mean distances and mean motions are subject only to periodical perturbations.

Soon after this Lagrange communicated a memoir to the Institute, in which he exhibited the application of the theory of the variations of arbitrary constants to all questions of mechanical science. This he did, by finding the variation of the six arbitrary constants, introduced by the integration of three differential equations of the second order (to which he had shown, in his Mécanique Analytique, that the motion of every system of bodies may be reduced), caused by a small disturbing force, in terms of the partial differential coëfficients of a quantity termed the disturbing function, with respect to the constants themselves, which enter implicitly into this function. Lagrange stated that this method might be applied to the determination of the motion of rotation of bodies about their centres of gravity, when the motion was disturbed by the attraction of a foreign body. This part of the theory was soon after supplied by Poisson in a memoir in which he investigated the expressions for the variation of the arbitrary constants, by a more direct analysis than that given by Lagrange.+

In the present state of analysis, as applied to celestial mechanics, the principal difficulty in computing the perturbations of the planetary motions consists in developing the disturbing function into a converging series, in functions of the

Calculus, p. 532.

Journal de l'Ecole Polytechnique, Vol. VIII., p. 266

et seq.

mean motions of the planets, and of the elements of their orbits. Several methods have been proposed for doing this, all of which involve an immense amount of labor when the development is carried to a high order of approximation. The analytical formulæ are so long and complicated, that there is great danger that errors will creep into them; and then the numerical application to all the planets becomes so tedious (and here, too, there is great danger of error), that but few have undertaken the labor necessary to carry the development to the higher powers of the eccentricities and inclinations. Among those who have performed this labor, we may mention M. Burchardt,* who carried the numerical calculations up to the sixth powers of the eccentricities. M. Binet carried the apарproximations up to the seventh powers of the eccentricities and inclinations, in a memoir presented to the Academy of Sciences in 1812. M. Pontécoulant subsequently repeated these calculations, and carried the approximations up to the sixth order with respect to the eccentricities and inclinations.† M. Leverrier has also carried the development of disturbing functions up to the seventh order of small quantities. +

Other methods have been employed for developing the disturbing function. The method of mechanical quadratures has been employed by several geometers. M. Poisson has published two papers on the subject; and M. Pontécoulant has treated the subject in his Theorie Analytique du Système du Monde. M. Hansen has also determined the coëfficients of the disturbing function by this method. Another method is to give particular values to the disturbing function, and then, by means of the equations formed between them and the corresponding series, eliminate as many coëfficients as is desirable, and thus determine their value. M. Leverrier has practised this method with complete success in computing the perturbations of Uranus by Saturn.T

* Memoires de l'Institute, 1808. + Système du Monde, tome iii., livre vi. Annales de l'Observatoire Imperiale, tome i. § Connaissance des Temps for the years 1825 and 1836. Tome iii.. livre vi., chap. ii. ¶ Con, des Temps for 1849.