Analysis of the Mccanlque Celeste of M. La Place. 3^ 



the different terms of which these formulae are composed, 

 and shows that we may easily extend them to the case in 

 which the number of the disturbing masses is given. 



The formulae v/hich give the disturbed motion contain- 

 ing in some of their terms the time without periodical signs, 

 the author employs the method described above to make 

 these terms disappear. This process leads to differential 

 equations among the constant arbitrary quantities of the 

 problem which are here the elements of the elliptic motion j 

 we obtain by this means the variations which these elements 

 undergo in consequence of the action of the disturbing 

 masses, at least so far as we neglect the second powers of 

 these masses, and those of the eccentricities and inclinations 

 of the orbits. The first property which this analysis dis- 

 plays, is, that the transverse axes of the orbits and the mean 

 motions are unalterable ; but the expression of the longitude 

 from which we draw this result being only approximate, it 

 is important to examine the nature of the terms which might 

 be there introduced by successive approximations ; for, if 

 there were any which were proportional to the square of the 

 time, the preceding property would ceas^ to take place, and 

 the transverse axes and Uie meaa motions might be indefi- 

 nitely altered. The axithor shows, that if we regard only the 

 first power of the disturbing masses, however far in other 

 jespccts we may extend the approximation with respect to 

 the eccentricities and inclinations, the expression of the 

 longitude will never contain any similar terms, at least 

 while the mean motions of the bodies of the system are in- 

 commensurable among each other, which is the case with 

 the solar system ; whence it results, that l)y confininn' our- 

 selves to the first powers of the disturbing masses, the trans- 

 verse axes of the orbits are constant, and the n)eau motions 

 uniform. The author afterwacds integrates the differen- 

 tial equations which determine the variations of the oilier 

 elliptic elements ; he discusses their extent, and shows that 

 the solar system can only oscillate around a mean state of 

 ellipticity and of inclination, from which it wKiidcrs little : 

 hence it follows, that the orbits of the planets ami sntcllites 

 never have been, and never will be, considerably eccentric^ 



C 4 nor 



