(65.) The bodies impinging were, in the last case, supposed to move in the same direction. We shall now consider the case in which they move in opposite directions. First, let the masses A and B be supposed to be equal, and moving in opposite directions, with the same velocity. Let C, _fig. 5._, be the point at which they meet. The equal motions in opposite directions will, in this case, destroy each other, and both masses will be reduced to a state of rest. Thus, the mass A loses all its motion in the direction A C, which it may be supposed to transfer to B at the moment of impact. But B having previously had an equal quantity of motion in the direction B C, will now have two equal motions impressed upon it, in directions immediately opposite; and these motions neutralising each other, the mass becomes quiescent. In this case, therefore, as in all the former examples, each body transfers to the other all the motion which it loses, consistently with the principle of “action and reaction.” The masses A and B being still supposed equal, let them move towards C with different velocities. Let A move with the velocity 10, and B with the velocity 6. Of the 10 parts of motion with which A is endued, 6 being transferred to B, will destroy the equal velocity 6, which B has in the direction B C. The bodies will then move together in the direction C B, the four remaining parts of A’s motion being equally distributed between them. Each body will, therefore, have two parts of A’s original motion, and 2 therefore will be their common velocity after impact. In this case, A loses 8 of the 10 parts of its motion in the direction A C. On the other hand, B loses the entire of its 6 parts of motion in the direction B C, and receives 2 parts in the direction A C. This is equivalent to receiving 8 parts of A’s motion in the direction A C. Thus, according to the law of “action and reaction,” B receives exactly what A loses. Finally, suppose that both the masses and velocities of A and B are unequal. Let the mass of A be 8, and its velocity 9: and let the mass of B be 6, and its velocity 5. The quantity of motion of A will be 72, and that of B, in the opposite direction, will be 30. Of the 72 parts of motion, which A has in the direction A C, 30 being transferred to B, will destroy all its 30 parts of motion in the direction B C, and the two masses will move in the direction C B, with the remaining 42 parts of motion, which will be equally distributed among their 14 component masses. Each component part will, therefore, receive 3 parts of motion; and accordingly 3 will be the common velocity of the united mass after impact.