OUTLINES OF NATURAL PHILOSOPHY. 



366. If the bodies are subjected at the same 

 time to the action of an accelerating force, the 

 sum made up by multiplying every body into the 

 square of its velocity, and adding all the products 

 together, is equal to the sum made up by multi- 

 plying every body into the square of its initial ve- 

 locity, and adding to the amount the sum made 

 up by multiplying every one of them into the 

 square of the velocity which it would have acqui- 

 red by the action of the accelerating force alone, 

 if it had moved freely in the same line which it 

 has actually described. 



a. See Dynamique par M. D'ALEMBERT ; where the 

 whole of the 4th chapter is employed in demon- 

 strating the different cases of these propositions. 

 CLAIRAULT has also given a demonstration, Mem. 

 Acad. des Sciences, 1742. DAN. BERNOUILLI had 

 before founded an entire system of Hydrodynamics 

 on the preservation of the vis viva, merely assuming 

 it as true. Hydrodynamica sive de Viribus et Moti- 

 bus Fluidorum, Argent. 1738. 



b. D'ALEMBERT has shewn, that the preservation of 

 the vis viva depends on this principle, That when 

 any number of forces are in equilibrium, the velo- 

 cities of the points to which the forces are applied, 

 estimated in the direction of the forces themselves, 

 are in the inverse ratio of those forces. Dyna- 

 mique, p. 267. edit. 2d. The conservation of the 



2 vis 



