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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 1lb. 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 1lb. 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.