472 Analysis of the Mccanique Cflestc of M. La Place. 



whose orbits have not yet been ascertained to be elliptic ) and 

 it follows from this, that, for each system of satellites, the 

 squares of the times of their .revolutions are as the cubes 

 of their mean distances from the centre of the planet: the 

 earth having but one satellite, we cannot apply this con- 

 sideration to it ; but the author shows that, if we determine 

 the lunar parallax according to the terrestrial experiments 

 upon gravity, and with the hypothesis that the recipro- 

 cal gravitation is as the square of the distance, the result 

 obtained by this way is perfectly conformable to the observa- 

 tions, whence it follows, that the attractive force of the earth 

 is the same as that of all the celestial bodies. These conclu- 

 sions give" rise to several important reflections, from which the 

 author infers this general consequence, that the particles 

 of matter attract each other in the direct ratio of the masses, 

 and the inverse ratio of the square of the distances. 



Conformably to this theory, the author establishes the 

 differential equations which determine the motion of a 

 system of bodies subjected to their mutual attraction, and 

 develops the small number of exact integrals which they 

 have hitherto been able to obtain : as observation only makes 

 us acquainted with the relative motions, he gives the fbfc» 

 mulae for the motion of a system of bodies subject to the 

 laws of gravitation round a body considered as the centre 

 of their motions, and develops the exact integrals which we 

 know how to deduce from them. In order to go further, 

 recourse must be had to the methods of approximation, 

 and we musUprofit by the facilities offered for this purpose 

 by the constitution of the system of the world : the author 

 shows that, according to this constitution, the satellites of 

 the planets are moved nearly as if they only obeyed the 

 action of the planet ; and the motion of the centre of gravity 

 of a planet, and of its satellites, is very nearly the same as 

 if each of these bodies was collected into its centre. He af- 

 terwards proceeds to inquire into the attractive properties 

 of spheroids, and establishes some general propositions 

 on this head, from which it follows that a poiyt placed in 

 the interior of a s pheric stratum is equally atttacted from all 

 parts, and that a point without the stratum is attracted by 



it 



