STABILITY OF THE SOLAU SYSTEM. 325 



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 Mecanique Celeste. 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 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 re- 

 searches on the secular variations of the planetary orbits. ( Mem. 

 Acacl 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. 

 ( Gmnaissance des Temps, 1843.) Translator. 



