[30] The results of Clairaut’s researches on the figure of the earth are mainly embodied in a remarkable theorem discovered by that geometer, and which may be enunciated thus:—­The sum of the fractions expressing the ellipticity and the increase of gravity at the pole is equal to two and a half times the fraction expressing the centrifugal force at the equator, the unit of force being represented by the force of gravity at the equator. This theorem is independent of any hypothesis with respect to the law of the densities of the successive strata of the earth.  Now the increase of gravity at the pole may be ascertained by means of observations with the pendulum in different latitudes.  Hence it is plain that Clairaut’s theorem furnishes a practical method for determining the value of the earth’s ellipticity.—­Translator.

[31] The researches on the secular variations of the eccentricities and inclinations of the planetary orbits depend upon the solution of an algebraic equation equal in degree to the number of planets whose mutual action is considered, and the coefficients of which involve the values of the masses of those bodies.  It may be shown that if the roots of this equation be equal or imaginary, the corresponding element, whether the eccentricity or the inclination, will increase indefinitely with the time in the case of each planet; but that if the roots, on the other hand, be real and unequal, the value of the element will oscillate in every instance within fixed limits.  Laplace proved by a general analysis, that the roots of the equation are real and unequal, whence it followed that neither the eccentricity nor the inclination will vary in any case to an indefinite extent.  But it still remained uncertain, whether the limits of oscillation were not in any instance so far apart that the variation of the element (whether the eccentricity or the inclination) might lead to a complete destruction of the existing physical condition of the planet.  Laplace, indeed, attempted to prove, by means of two well-known theorems relative to the eccentricities and inclinations of the planetary orbits, that if those elements were once small, they would always remain so, provided the planets all revolved around the sun in one common direction and their masses were inconsiderable.  It is to these theorems that M. Arago manifestly alludes in the text.  Le Verrier and others have, however, remarked that they are inadequate to assure the permanence of the existing physical condition of several of the planets.  In order to arrive at a definitive conclusion on this subject, it is indispensable to have recourse to the actual solution of the algebraic equation above referred to.  This was the course adopted by the illustrious Lagrange in his researches on the secular variations of the planetary orbits. (Mem.  Acad.  Berlin, 1783-4.) Having investigated the values of the masses of the planets, he then determined, by an approximate