Equilibrium between Forces directly opposite. 1 9 



?i, might also be given for supposing n to prevail over ra, since 

 they are by hypothesis in all respects equal. 



33. Let us suppose now, that ra is double of n, but that i>, 

 at the same time is double of w, that is, that n passes over two 

 feet, for example, in a second, while m passes over one foot in a 

 second. It is clear that we may consider m as composed of two 

 masses equal each to n ; and that at the instant of meeting, we 

 may represent the body n as having a velocity of one foot in a 

 second, to which is added at the same instant, another velocity of 

 one foot in a second. We may then conceive, that in meeting 

 the mass n expends one of its velocities against a portion of the 

 mass m equal to itself, and its other velocity against the remain- 

 ing portion of m of the same magnitude. 



If now, instead of supposing the masses m and n in the ratio 

 of 2 to 1, and their velocities in the ratio of 1 to 2, we suppose 

 them in any other ratio, it is evident that we may always con- 

 ceive the greater mass as decomposed into a certain number of 

 portions equal each to the smaller, and of which each shall des- 

 troy in the smaller, a velocity equal to its own. We may there- 

 fore consider the following proposition as established. 



Two bodies which act directly against each other in the same 

 straight /me, are in equilibrium when their quantities of motion are 

 equal; that is, when the product of the mass of the one into the 

 velocity with which it moves, or tends to move, is equal to the 

 product of the mass of the other into its actual or virtual velocity. 



This proposition is to be regarded as general, whether the 

 two bodies move freely and directly the one against the other, 

 or whether they act against each other by the intervention of a 

 rod inflexible and without mass, or whether they are considered 

 as pulling in opposite directions by means of a thread m n inca- 

 pable of being extended. And reciprocally, if two bodies are in 

 equilibrium, we may conclude that their motions are directly 

 opposite, and that their quantities of motion are equal. 



34. We infer, moreover, that if three or a greater number of 

 bodies ra, n, o, &c., moving, or tending to move, in the same Fig. a, 

 straight line, with velocities w, v, zy, &c., are in equilibrium, the sum 



