(55.) If a mass A, _fig._ 4., moving towards C, impinge upon an equal mass, which is quiescent at B, the two masses will move together towards C after the impact. But it will be observed, that their speed after the impact will be only half that of A before it. Thus, after the impact, A loses half its velocity; and B, which was before quiescent, receives exactly this amount of motion. It appears, therefore, in this case, that B receives exactly as much motion as A loses: so that the real quantity of motion from B to C is the same as the quantity of motion from A to B. Now, suppose that B consisted of two masses, each equal to A, it would be found that in this case the velocity of the triple mass after impact would be one-third of the velocity from A to B. Thus, after impact, A loses two-thirds of its velocity and, B consisting of two masses each equal to A, each of these two receives one-third of A’s motion; so that the whole motion received by B is two-thirds of the motion of A before impact. By the impact, therefore, exactly as much motion is received by B as is lost by A. A similar result will be obtained, whatever proportion may subsist between the masses A and B. Suppose B to be ten times A; then the whole motion of A must, after the impact, be distributed among the parts of the united masses of A and B: but these united masses are, in this case, eleven times the mass of A. Now, as they all move with a common motion, it follows that A’s former motion must be equally distributed among them; so that each part shall have an eleventh part of it. Therefore the velocity after impact will be the eleventh part of the velocity of A before it. Thus A loses by the impact ten-eleventh parts of its motion, which are precisely what B receives. Again, if the masses of A and B be 5 and 7, then the united mass after impact will be 12. The motion of A before impact will be equally distributed between these twelve parts, so that each part will have a twelfth of it; but five of these parts belong to the mass A, and seven to B. Hence B will receive seven-twelfths, while A retains five-twelfths. (56.) In general, therefore, when a mass A in motion impinges on a mass B at rest, to find the motion of the united mass after impact, “divide the whole motion of A into as many equal parts as there are equal component masses in A and B together, and then B will receive by the impact as many parts of this motion as it has equal component masses.”