(59.) In the cases of collision of which we have spoken, one of the masses B was supposed to be quiescent before the impact. We shall now suppose it to be moving in the same direction as A, that is, towards C, but with a less velocity, so that A shall overtake it, and impinge upon it. After the impact, the two masses will move towards C with a common velocity, the amount of which we now propose to determine. If the masses A and B be equal, then their motions or velocities added together must be the motion of the united mass after impact, since no motion can either be created or destroyed by that event. But as A and B move with a common motion, this sum must be equally distributed between them, and therefore each will move with a velocity equal to half the sum of their velocities before the impact. Thus, if A have the velocity 7, and B have 5, the velocity of the united mass after impact is 6, being the half of 12, the sum of 7 and 5. If A and B be not equal, suppose them divided into equal component parts, and let A consist of 8, and B of 6, equal masses: let the velocity of A be 17, so that the motion of each of the 8 parts being 17, the motion of the whole will be 136. In the same manner, let the velocity of B be 10, the motion of each part being 10, the whole motion of the 6 parts will be 60. The sum of the two motions, therefore, towards C is 196; and since none of this can be lost by the impact, nor any motion added to it, this must also be the whole motion of the united masses after impact. Being equally distributed among the 14 component parts of which these united masses consist, each part will have a fourteenth of the whole motion. Hence, 196 being divided by 14, we obtain the quotient 14, which is the velocity with which the whole moves. (60.) In general, therefore, when two masses moving in the same direction impinge one upon the other, and after impact move together, their common velocity may be determined by the following rule: “Express the masses and velocities by numbers in the usual way, and multiply the numbers expressing the masses by the numbers which express the velocities; the two products thus obtained being added together, and their sum divided by the sum of the numbers expressing the masses, the quotient will be the number expressing the required velocity.”