40 Analysis of the Mecanique Celeste of M. La Place. 



nor inclined lo each other, at least if we regard only the 

 mutual action of these bodies. The variations determined 

 by the preceding analysis, taking place with great slowness, 

 have been named secular ; and we may, during a long iii- 

 ttrval, suppose them proportional to the time: the author 

 jrives the method of obtaining them under that form, which 

 is useful for astronomical purposes. 



These inquiries make known some relations among the 

 elements of the orbits which are only approximate : the au- 

 thor develops those which take place in general, whatever 

 may be the eccentricities and the inclinations : he afterwards 

 gives the necessary formulae for determining, with respect 

 to the solar system, the position of the invariable plane upon 

 which the sum of the areas described by the projections of 

 the radii vectores of the bodies of the system, multiplied re- 

 spectively by the masses of these bodies, is a maximuni. 

 The research of this plane becomes very important, on ac- 

 count of the proper motions of the stars and of the ecliptic ; 

 bu|t it requires that we should know the masses of tlie co- 

 mets, and the elements of their orbits : happily, these 

 masses appearing to be very small, we may, without any 

 perceptible error, neglect their action on the planets : con- 

 sidering then the motion of two orbits inclined one to the 

 other by anv given angle, the author shows that, indepen- 

 dently of every extraneour cause, the two orbits will always 

 cut the invariable plane relative to their system in the same 

 straight line, the ascending noHe of the one comciding with 

 the descending node of the other ; and he gives, upon the 

 supposition of very small inclinations, the expression of the 

 motion of this intersection. 



The preceding method only giving the inequalities inde- 

 pendent of the mutual configuration of the bodies of the 

 system, the author resumes this problem by a different pro- 

 cess ; he infers from analytical considerations detailed above, 

 that the disturbed motion of the celestial bodies may be re- 

 ferred to the laws of elliptic motion, supposing the elements 

 of this motion variable ; he shows that these results may also 

 be drawn immediately from the consideration of elliptic mo- 

 tion, regarding the disturbed planet, as oscillating in a very 



small 



