For if we interconnect the balls by loose elastic threads, and then give, as before, a transverse or sideways impulse to the first ball, this will pull the second one and set it swinging, but it will be pulled back itself, and will be to some extent deprived of its motion. The same sharing or division of energy will take place between the second and third, and third and fourth balls, and so on. Hence the initial solitary vibration of the first ball draws out into a wave-train, and the originally imparted energy is spread out over a number of balls, and not concentrated in one of them. Accordingly, as time goes on, the wave-train is ever extending in length and the oscillatory motion of each ball is dying away, and the original energy gets spread over a wider and wider area or number of balls, but is propagated with less speed than the wave-velocity for that medium. There need be no difficulty in distinguishing between the notion of a wave-velocity and a wave-train velocity, if we remember that the wave travels a distance equal to a wave-length in the time taken by one oscillation. Hence the wave-velocity is measured by taking the quotient of the wave-length by the time of one complete vibration. If, for example, the wave-length of a water wave is 4 inches, and we observe that twelve waves pass any given point in 3 seconds, we can at once infer that the wave-velocity is 16 inches per second. The transference of energy may, however, take place so that the whole group of waves moves forward much more slowly. They move forward because the waves are dying out in the rear of the group and being created in the front, and the rate of movement of the group is, in the case of deep-water waves, equal to half that of the single-wave velocity. A very rough illustration of this difference between a group velocity and an individual velocity may be given by supposing a barge to be slowly towed along a river. Let a group of boys run along the barge, dive over the bows, and reappear at the stern and climb in again. Then the velocity of the group of boys on the barge is the same as the speed of the barge, but the speed of each individual boy in space is equal to the speed of the barge added to the speed of each boy relatively to the barge. If the barge is being towed at 3 miles an hour, and the boys run along the boat also at 3 miles an hour, then the velocity of the group of boys is only half that of the individual boy, because the former is 3 miles an hour and the latter is 6 miles an hour.