8. We will next examine more closely the separate movements of particles at the surface of water, and their joint result the transmission of waves. The brothers Heinrich and Wilhelm Weber published, in 1825, a series of very careful observations on water-waves passing along troughs with glass sides not unlike very long and very narrow aquariums. By ingeniously planned arrangements, which it would take too long to describe here, they ascertained that each particle on the surface continued, during the passage of a succession of equal waves, to describe a fixed oval path in a vertical plane, the longer diameter of the oval being horizontal. From the fact that the difference between the horizontal and vertical diameters of the oval diminished as the depth of the water increased, the brothers Weber inferred that at the surface of very deep water these diameters would be equal, and the orbits of the particles therefore exact circles.

Fig. 5 shows how a series of particles, by describing fixed circular paths in a vertical plane, cause progressive waves to be transmitted horizontally. At the top of the figure are 17 spots representing as many particles floating at equal distances along a straight line in the undisturbed surface of very deep water indicated by dots in the

figure. The equal vertical circles are the paths which the particles are respectively about to describe in the direction in which the hands of a watch move, as indicated by a curved arrow in the figure, and with equal uniform velocities.

(0) represents the state of things at a movement when the group of particles plot out two complete equal waves included by the brackets A and B, each consisting of a crest to the right and trough to the left as marked by the sub-brackets a, a' and b, b′ respectively. The horizontal dotted line represents, as before, the section of the undisturbed surface or level-line.. The reader will at once observe, on looking along the line of circles from left to right, that each particle occupies in its own orbit a position one-eighth of a revolution behind that of the particle to the left of it.

(1) shows the arrangement of the particles when an interval of time equal to one-eighth of the period of a complete circle-revolution has elapsed from the moment at which they were grouped as in (0).

Similarly (2) represents the state of things after two-eighths of that period, (3) after three-eighths of it, and so on, until in (8), one complete period of revolution having in all elapsed, the particles are necessarily again in the same positions as those shown in group (0).

If now the reader allows his eye to be guided by the oblique dotted lines of the figure, he will readily perceive that the wave A has moved horizontally to the right at each of the above stages and, after the lapse of one complete period of particle-revolution occupies, in (8), the position which the wave B held in (0), i.e. has traversed its own wave-length from left to right. We may express this result generally by saying that while an individual particle performs one complete orbital revolution, the wave advances one wave-length. This is the fundamental proposition of wave-motion, and should be carefully mastered and remembered. In the figure employed to demonstrate it a wave has been only roughly plotted out by a small number of spots, and its movements estimated at but a few arbitrarily chosen instants. By increasing the numbers of these two elements, however, we might make an indefinitely near approach to continuity both of form and of motion.

9. The characteristic phenomenon of wavemotion, viz. an apparent forward movement unshared by the materials which give rise to it, though most frequently seen on the surface of water, is by no means confined to fluid bodies. When a carpet is being shaken, bulging forms exactly like water-waves are seen running along it. A flexible string, one end of which is tied to a fixed point and the other

held in the hand, exhibits the same phenomenon when the loose end is sharply jerked aside. It has accordingly been found convenient to extend the term wave in order to designate as wave-motion any movement coming under the definition just given. We proceed to an instance in which the individual particles, instead of describing circular orbits move to-and-fro, i.e. vibrate, in straight lines perpendicular to the direction of wave-propagation.

Before applying to this case the method followed in Fig. 5 it will be desirable to define what is meant by a 'complete vibration' as that term is used by English writers.

Let a point move from A' to A along the straight

Fig.6

line A'A and then back again to A'. A French writer would describe this movement as made up of two complete vibrations executed in opposite directions. An English writer would designate it as one complete vibration, consisting of two halfvibrations performed in opposite directions. This latter usage will be adhered to in the sequel. The