HISTORY OF PHYSICAL ASTRONOMY.

Ill

distance as early as 1770, and Laplace subsequently obtained similar ex-I pressions relative to the eccentricity, the inclination, and the longitude■ of the node*. The importance of Lagrange’s researches on the presentoccasion was, however, of a twofold character, for, besides exhibiting thevariations of all the elements under the same form, the direct analysisj by which he obtained his results, and the general aspect under which hei viewed the elements (considering them merely as the six arbitrary con-stants involved in the complete integrals of the differential equations,j without reference to their individual functions in an elliptic orbit), seemed' to indicate a general theory, of which the variations of the elements of the; planetary orbits served only as a particular illustration.

On the very day that Lagrange communicated to Laplace the interesting result of his researches on the variations of the elements, the latterexhibited to him analogous expressions, to which he had just been con-ducted by an investigation of the same subject. Such was the remark-able parity of inventive power which distinguished the rival efforts ofthese illustrious geometers, and which so long held the judgment of Eu-rope in suspense upon the question of their relative merits!

Notwithstanding the identity of the results obtained by Lagrange andLaplace on the present occasion, it was not difficult to discern in the re-searches of each geometer the peculiar bent of his genius. Lagrange,who wielded the powers of the transcendental analysis with unrivalledeffect, considered the subject in its most abstract form, and by this meansimparted a generality to his researches, which could not fail to lead toimportant results. Laplace, always less enamoured with the beauty ofmathematical speculation than he was anxious to unfold the system of theworld, contented himself with reducing the expressions for the variationsof the epoch and perihelion to the same form with those relating to theother elements, not considering them in connexion with any general prin-ciple, hut simply deriving them from established formulae. In effectingthis object, he was considerably aided by a relation between the two ele-ments which Poisson had recently discovered. These results appeared ina supplement to the third volume of the Mecanique Celeste f. Laplace ex-hibited a beautiful illustration of their practical utility, by employing themin the investigation of the lunar inequalities occasioned by the spheroidalfigure of the earth. The expressions which he obtained by this processfor the inequalities both in longitude and latitude coincided exactly, inall respects, with those he had already arrived at by the usual method ofapproximation. It is exceedinglj' interesting to trace the final agreementof two methods differing so essentially in their original conception.

Soon after Lagrange and Laplace arrived at the above-mentioned results,llie former of these geometers communicated a memoir to the Institute , inwhich he exhibited the application of the theory of the variation of arbi-trary constants to all questions of mechanical science). Assuming asthe basis of his researches the three differential equations of the secondorder to which he had already shewn, in the Mecanique Analytique , thattlie motion of every system of bodies might be reduced, he supposed thesix arbitrary constants of the complete integrals to vary from the effect ofa small disturbing force and to be represented in the differential equations

* Mec. Cel., liv. ii. chap. viii.

+ Laplace first announced them at the sitting of the Bureau des Longitudes , 17tli ofAugust, 1808.

j Memoires de l’lnstitut, P. 237, et seq.