Analysis of the Mecarnquc Celeste of M. La Place. 2&T 



tion which should exist between the forces which solicit the 

 system in order to fulfil this condition, and he makes appli- 

 cation of it lo the equilibrium of a homogeneous fluid mass 

 covering a fixed solid nucleus, and of a given figure. He 

 afterwards gives the general equation of the movement of 

 any system of bodies, which he deduces from that of equili- 

 brium ; and he draws from it the principles of the preser- 

 vation of living forces, of areas, of the motion of the centre 

 of gravity, and of the least action. He fixes the circum- 

 stances in which these principles take effect, and gives fhe 

 method of estimating the alteration which that of living 

 forces undergoes in the sudden changes of ihe motion of the 

 system. In treating of the. principle of the areas, he shows 

 that in the motion ot a system of bodies animated solely by 

 their mutual attraction, <md by forces directed towards the 

 origin of the coordinates, there exists a plane passing by this 

 origin, and which enjoys the following' remarkable proper- 

 ties : 1st, The sum of the areas traced upon this plane by 

 the projections of the vector radii of the bodies, and multi- 

 plied respectively by their masses, is here the greatest pos- 

 sible. 2dIy,This same sum is null upon all the planes which 

 arc perpendicular to it ; the principles of its living forces 

 and of the areas, still taking place with respect to the centre 

 of gravity, even supposing it to have an uniform and recti- 

 linear movement. Hence it results that we may determine a 

 plane passing by this moveable origin, and upon which the 

 sum of the areas described by the projections of the vector 

 radii of bodies, and multiplied respectively by their masses, 

 is the greatest possible. The author shows that this plane 

 is parallel to that which passes by the fixed origin, and sa- 

 tisfies the same conditions. Hence he infers, that the plane 

 passing by the centre of gravity, and determined according to 

 the preceding conditions, always remains parallel to itself in 

 the movement of the system; a singular ad vantage, and which 

 renders it of the greatest utility. It is another remarkable 

 circumstance, that every plane parallel to the. above, and 

 passing by any one of the bodies of the system, will enjoy 

 analogous properties. After having obtained these valuable 

 results, the author examines the laws of movement which 

 . r could 



