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 solution, the values of the several
roots of the algebraic equation upon which the variations of the
eccentricities and inclinati