474 Analysis of the Mecanique Celeste ofM. La Place, 



partial differences, has the advantage of giving the arbitrary 

 quantities in functions of the co-ordinates, and of their first 

 differences, which is frequently useful : the author deduces 

 from it the relations which take place between these arbi- 

 trary quantities, and the elements which determine the na- 

 ture of the conic section and its position in space : finally, 

 he integrates the differential equation which gives the time 

 in a function of the radius vector; and the motion of two 

 bodies is thus determined by three equations, between the 

 eccentric anomaly, the true anomaly, the mean anomaly, 

 and the radius vector of the orbit : these equations being of 

 a nature not capable of being resolved except by approxima- 

 tion, the author details some general theorems upon the re- 

 duction of functions into series; and applying these results 

 to the elliptical motion of the planets, he deduces from them 

 the values of the eccentric anomaly, the true anomaly, and 

 of the radius vector, in convergent series of the sines and co- 

 sines of the mean anomaly: by referring the motion of the 

 planet to a fixed plane a little inclined to that of the orbit, 

 these series furnish the means of determining by approxi- 

 mation the latitude and longitude of the planet with respect 

 to the fixed plane, as well as the projection of the radius of 

 the orbit upon the same plane. The author explains the the- 

 ory of motion in a very eccentric ellipsis, and thence de- 

 duces the theory of the parabolic motion applicable to co- 

 mets : he afterwards considers the hyperbolic motion ; and 

 then arriving at Kepler's law, according to which the 

 squares of the revolutions of different planets are to each 

 other as the cubes of the transverse axes of their orbits, he 

 shows that this law is not accurately true, and that it only 

 takes place when we neglect the action of the planets upon 

 each other, and upon the sun, and when we consider their 

 masses as infinitely small with respect to that of the sun. 

 He shows the use of these results in determining the ratios 

 of the masses of the planets which have satellites, to the 

 mass of the sun. v 



After having'detailed the theory of elliptic motion, and 

 the method of calculating it by converging series in the two 

 cases of Nature, that of orbits almost circular, and that of 



orbits 



