Let _a_ express the velocity of A. Let _b_ express the velocity of B. Let _x_ express the velocity of the united masses of A and B after impact, each of these velocities being expressed in feet per second, and the masses of A and B being expressed by the weight in pounds. We shall then have the momenta or moving forces of A and B before impact, expressed by A × _a_ and B × _b_, and the moving force of the united mass after impact will be expressed by (A + B) × _x_. The moving force of A after impact is A × _x_, and therefore the force it loses by the collision will be (A × _a_ - A × _x_). The force of B after impact will be B × _x_, and therefore the force it gains will be B × _x_ - B × _b_. But since the force lost by A must be equal to the force gained by B, we shall have A × _a_ - A × _x_ = B × _x_ - B × _b_ from which it is easy to infer (A + B) × _x_ = A × _a_ + B × _b_ and if it be required to express the velocity of the united masses after impact, we have _x_ = (A × _a_ + B × _b_)/(A + B) When it is said that A × _a_ and B × _b_ express the moving forces of A and B, it must be understood that the _unit_ of momentum or moving force is in the case here supposed, the force with which a mass of matter weighing 1 lb. would move if its velocity were 1 foot per second, and accordingly the forces with which A and B move before impact are as many times this as there are units respectively in the numbers signified by the general symbols A × _a_ and B × _b_. In like manner, the force of the united masses after impact is as many times greater than that of 1 lb. moving through 1 foot per second as there are units in the numbers expressed by (A + B) × _x_. (64.) These phenomena present an example of a law deduced from the property of inertia, and generally expressed thus--“action and reaction are equal, and in contrary directions.” The student must, however, be cautious not to receive these terms in their ordinary acceptation. After the full explanation of inertia given in the last chapter, it is, perhaps, scarcely necessary here to repeat, that in the phenomena manifested by the motion of two bodies, there can be neither “action” nor “reaction,” properly so called. The bodies are absolutely incapable either of action or resistance. The sense in which these words must be received, as used in the _law_, is merely an expression of the _transfer_ of a certain quantity of motion from one body to another, which is called an _action_ in the body which loses the motion, and a _reaction_ in the body which receives it. The _accession_ of motion to the latter is said to proceed from the _action_ of the former; and the _loss_ of the same motion in the former is ascribed to the _reaction_ of the latter. The whole phraseology is, however, most objectionable and unphilosophical, and is calculated to create wrong notions.