1820.] Roi/al Academy of Sciences. 223 



from observations to that of the physical cause of the phenomena. 

 He saw that this equaHty of the areas necessarily shows the 

 power that retains the planet in its orbit to be directed towards 

 the sun. Thus every one of Kepler's laws became a theorem in 

 dynamics. D'Arcy, Bernouilh, and Euler, well knew that if the 

 areas were drawn upon any plane, the sum of the areas, measured 

 in one direction, would augment in proportion to the time 

 elapsed." 



Here the Committee mention in an abridged manner the theo- 

 rem of Newton relating to the conservation of the centre of 

 gravity ; that of M. De Laplace concerning the plane of the 

 maximum of <^he areas, the researches of Euler concerning the 

 measure and composition of the momenta. " The propositions 

 relative to this composition, and many new theorems concerning 

 the same subject, are expressed in the clearest and the most 

 elegant manner in the Treatise, and the Memoirs which M. 

 Poinsot has published upon Statics. Every one of these results 

 are to be found in them, deduced by a plain method pecuhar to 

 him, which has the advantage of rendering them easy to^ be 

 perceived, and of proving, in a direct manner, that the forces 

 of rotation are decomposed, distributed, and destroyed, accord- 

 ing to rules entirely similar to those which concur in the forces 

 of translation. The exposition of the properties relative to areas, 

 which Laplace has given in his Mecanique Analytique, ought to 

 be added to this enumeration. 



" M. Binet's method consists in deducing from the differen- 

 tial equations of the motion, the expressions relative to the areas 

 produced, and to their fluxions of the first and second order, by 

 proving that the expressions combine among themselves in the 

 same manner as those of the arcs described by the moving body, 

 and those of the velocities. This analogy between the areas and 

 the trajectories may be considered in another point of view; in 

 reality, if we suppose that, in the general equation which shows 

 the sum of the projected areas to mcrease in proportion to the 

 times, the centre of the radii vectores is placed at an infinite 

 distance from the origin of the coordinates, we shall see directly 

 that the velocity with which the sum of the areas increase is the 

 velocity with which the centre of gravity of the system departs 

 from a fixed plane. And in this manner the theorem concerning 

 the motion of the centre of gravity is deducible from that of the 

 preservation of the areas. The case is the same in regard to the 

 equation which expresses the three parts of the vis viva of 

 rotation. 



'• The latter end of the memoir presents an ingenious collection 

 of several general theorems of mechanics. In order to show 

 that these propositions arise from one common source, the 

 author adds to them the differential equations of the motions, 

 multiplied by the coefficients which the variable quantities may 

 contain, and their differentials of the first order. He proposes to 



