STABILITY OF THE SOLAR SYSTEM. 825 
_ Although the invariability of the mean distances of the 
planetary orbits has been more completely demonstrated 
since the appearance of the memoir above referred to, 
that is to say by pushing the analytical approximations 
to a greater extent, it will, notwithstanding, always con- 
stitute one of the admirable discoveries of the author of 
the Mécanique Céleste. Dates, in the case of such sub- 
roots of this equation be equal or imaginary, the corresponding ele- 
ment, whether the eccentricity or the inclination, will increase indefi- 
nitely 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 exist- 
ing physical condition of the planet. Laplace, indeed, attempted to 
prove, by means of two well-known theorems relative to the eccen- 
tricities 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 ata 
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 re- 
searches 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 inclinations of the orbits depended. 
In this way, he found the limiting values of the eccentricity and in- 
clination for the orbit of each of the principal planets of the system. 
The results obtained by that great geometer have been mainly con- 
firmed by the recent researches of Le Verrier on the same subject. 
( Connaissance des Temps, 1848.)— Translator. 
