Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/826

760 thesis concerning the interior constitution of the earth. In 1754 he published the first two volumes of his Re searches on the System of the World. In this work he applied the formulae by which he had calculated the motions of the moon to the motions of the planets disturbed by their mutual attraction, and pointed out the simplest method of determining the perturbations of the motions of a planet occasioned by the action of its own satellites. D Alembert also treated the subject of the figure of the earth in a much more general manner than had been done by Clairaut, who had confined his investigations to the case of a spheroid of revolution. He determined the attraction of a spheroid of small eccentricity, whose surface can be represented by an algebraic equation of any order whatever, even supposing the spheroid to be composed of strata of different densities.

The first memoir of on the planetary perturbations was transmitted to the secretary of the Academy of Sciences in July 1747, some months before Clairaut and D Alem- bert had communicated their solutions of the problem of three bodies, and it carried off the prize which the academy had offered for the analytical theory of the motions of Jupiter and Saturn. In this memoir Euler gave the differ ential equations of the elements of the disturbed planet, but withheld the analysis by which he had been con ducted to them. This analysis, however, he subsequently expanded in two memoirs, the first of which appeared in the Berlin Memoirs in 1749, and the second in those of St Petersburg in 1750. Of these supplementary memoirs the first is remarkable on several accounts. It contains the first example of a method which has been fruitful of important consequences namely, that of the variation of the arbitrary constants in differential equations, and the development of the radical quantity which expresses the distance between two planets in a series of angles, multi ples of the elongations. The expressions which he gave for the several terms of this series were simple and elegant ; and ho demonstrated a curious relation subsisting among any three consecutive terms, by means of which all the terms of the series may be calculated from the first two. He was thus enabled to develop the perturbing forces in terms of the sines and cosines of angles increasing with the time, and thereby to surmount a very great analytical difficulty. Notwithstanding, however, the great merit of Euler s memoir, several of the formulae expressing the secular and periodic inequalities were found to be inac curate ; and in order to procure a correction of these errors, and give greater perfection to so important a theory, the academy again proposed the same subject for the prize of 1752. This prize Avas also carried off by Euler. In the memoir which he presented on this occasion, he con sidered simultaneously the motions of Jupiter and Saturn, and determined, in the first instance, the amount of their various inequalities, independently of the consideration of the eccentricities of their orbits. Pushing the approxima tions farther, and having regard to the inequalities depend ing on the eccentricities, he arrived at a most important result relative to the periodic nature of the inequalities occasioned by the mutual perturbations of the planets ; which laid the foundation of the subsequent discovery by Lagrange and Laplace of the permanent stability of the planetary system. He demonstrated that the eccentricities and places of the apheliaof Jupiter and Saturn are subject to constant variation, which is confined, however, within certain fixed limits ; and he computed that the elements of the orbits of the two planets recover their original values after a lapse of about 30,000 years. In the year 175G the Academy of Sciences crowned a third memoir of Eubr on the same subject as the two former, namely, the inequalities of the motions of the planets produced by their reciprocal attractions. This memoir analytically considered is also of great value. The method which he followed and illustrated has since been generally adopted in researches of the same nature, and consists in regarding as variable, in consequence of the disturbing forces, the six elements of the elliptic motion, viz., 1st, the major axis of the orbit, which, by the law of Kepler, gives the ratio of the differen tial of the mean longitude to the element of the time ; 2c/, the epoch of this longitude ; 3d, the eccentricity of the orbit; 4th, the motion of the aphelion; 5th, the inclina tion of the orbit to a given fixed plane ; and, Gt/t, the longitude of the node. By considering separately the variations introduced into each of these elements by the disturbing forces, Euler obtained some important results ; but even in this memoir his theory was not rendered com plete. He did not consider the variation of the epoch ; and the expression which he gave for the motion of the aphelion did not include that part of it which depends on the ratio of the eccentricities of the orbits of the disturbed and disturbing planet. Besides, the third memoir, like the two former, contained several errors of computation, which, by leading to results known to be wrong, probably prevented the author himself from being aware of the full value of the ingenious methods of procedure which he hail described. Euler concluded this important memoir by mak ing an extended application of his formulae to the orbit of the earth as disturbed by the action of the planets. From some probable suppositions, first employed by Newton, relative to the ratios of the masses of the planets to that of the sun, he determined the variation of the obliquity of the ecliptic at 48&quot; in a century, a result which agrees well with observation. By this determination the secular variation of the obliquity of the ecliptic, which had been regarded by Lahire, Lemonnier, D Alembert, and other eminent astronomers as uncertain, was placed beyond doubt. The three memoirs which we have mentioned con tain the principal part of Euler s labours on the perturba tions ; but physical astronomy is indebted to him for many other researches. He gave a solution of the problem of the precession of the equinoxes, and made several important steps in the lunar theory, with Avhich he seems to have occupied himself before he undertook the investigation of the planetary perturbations. In the year 1772, when entirely blind, he directed his son, Albert Euler, and two illustrious pupils, Krafft and Lexell, in the composition of a work of enormous labour on the same subject, which was undertaken with a view to discover the cause of the moon s acceleration. This work was concluded with a set of lunar tables deduced entirely from theory ; but they were found to be far inferior to those of Mayer, and in some respects hardly equal to those of Clairaut.

The first theory of Euler formed the basis of the excellent lunar tables which were calculated by Tobias Mayer, and first published in the Memoirs of the Academy of Gottingen in 1753. Mayer was a skilful astronomer, and determined the co-efficients of the arguments of the different lunar inequalities from his own observations. He continued to correct and improve his tables till the time of his death, which happened in 1762, when a copy of them, containing his last corrections, was presented by his widow to the Board of Longitude in London. They were printed along with the author s lunar theory in 1765. Subse quently, the Board of Longitude directed Mason, who had been assistant to Bradley, to revise them, under the super intendence of Dr Maskelyne. Mason compared them with about 1200 of Bradley s observations, corrected the co efficients of Mayer, and introduced some new equations which had been indicated by that astronomer, but which he had considered as too uncertain, or of too small a value, to render it necessary to load his tables with them. 