Analysis of the Mecanique Celeste of M> La Place. 473 



it as if its mass were entirely collected to its centre ; pro- 

 perties which also take place with respect to globes formed 

 of concentric layers, of density variable from the centre to 

 the circumference: the author inquires what are, the laws of 

 attraction in which these effects subsist ; and he proves that, 

 among the infinite number of laws which render the attrac- 

 tion very small at great distances, the law of Nature is the 

 only one in which a spheric stratum attracts a poinjt placed 

 without it, as if it was all collected to its centre: he proves 

 also, that this law is the only one in which the action of the 

 layer upon a point placed within it is nothing : he also makes 

 a second application of the same formulae, to the case in 

 which the attracting body is a cylinder whose base is a re- 

 entering curve, the length of which is infinite; he demon- 

 strates that, when this curve is a circle, the action of the cy- 

 linder upon a point without it, is reciprocally as, the distance 

 from its axis to this point; and that> if the attracted point is 

 situated in the interior of a circular cylindric layer of a 

 constant thickness, it is equally attracted from all parts. 

 The formulas of the motion of a body give rise to some 

 very remarkable conditional equations : the author develops 

 them, and points out their use for verifying the calculations 

 of the theory, and the theory itself of universal gravity ; after 

 which he presents the various transformations which it may 

 be most frequently useful to subject the differential equa- 

 tions to, of the motion of any system of bodies animated by 

 their mutual attraction. The bodies which compose the so- 

 lar system, moving nearly as if they obeyed only the prin- 

 cipal force which animates them, and the perturbat'ing forces 

 not being very considerable, the author previously gives as 

 a first approximation, the exact determination of the motion 

 of two bodies which attract each other directly in the ratio of 

 the masses, and inversely as the square of the distances : he 

 explains successively three different methods of integrating 

 differential equations relative to this hypothesis: the second 

 of these methods is founded upon an elegant theorem rela- 

 tive to the integration of differential equations of the first 

 degree, and of any order whatever. The third, which makes 

 t\\e required integrals of one equation only to depend on 



partial 



