completion of a vibration is thus to be determined by the moving point having traversed its entire path, in both directions, and this is equally true if instead of starting from either extremity it occupies initially an intermediate position such as O and moves from 0 to A, from A to A' and from A' to O; or reversely from O to A', from A' to A and from A to O.

The 'period' of a vibration accordingly means the time which it takes the moving point to perform one such complete cycle of its motion.

Let the 17 spots lying equidistantly along the dotted straight line at the head of Fig. 7 represent as many particles of an elastic string stretched between fixed points of attachment not shown in the figure. These particles are about to vibrate in straight lines perpendicular to the direction of the string. (0) shows the state of things when the particles plot out two complete waves A and B. The distinction between 'crest' and 'trough' has now disappeared. All that can be said is that each wave is formed of two protuberances lying on opposite sides of the undisturbed position of the string, which replaces the 'level-line' of the water-waves and is, like it, represented by a dotted line in the figure.

By looking along 0 from left to right the reader

will notice that each particle occupies in its own path a position one-eighth of a period of vibration behind that of the particle next to the left of it. (Compare p. 16.) (1), (2), (3)...show the positions of the particles after one-eighth, two-eighths, three-eighths... of a complete period of particle-vibration. By following any one of the vertical lines of spots it will be seen that in the instance selected for this figure, each particle moves more rapidly in the neighbourhood of its undisturbed position than it does near the extremities of its swing. In (8) the particles have returned to their original positions and the wave A is where B was in (0). The particles have completed one vibrational cycle and the wave has advanced by its own length. This result may be thus generalised : While an individual particle performs one complete vibration the wave advances one wave-length. The proposition proved above (p. 17) for water-waves is, therefore, also true of waves due to transverse vibrations, i.e. such as are executed perpendicularly to the direction of wave-propagation.

As waves thus produced are of leading importance in the theory of Sound, it is necessary to study them in some detail.

Let a particle originally at rest at O in the initial line (Fig. 7 bis) be cooperating in the transmission This wave is drawn in the figure in two

of a wave.

positions such that the two points of its curve the most distant from the initial line, A and A', are

Fig. 7 (bis)

A

situated in two straight lines OA and OA' drawn through O in opposite directions, each perpendicular to the initial line. It is evident that, at the moments when the wave is in these positions, the particle originally at O will be at A and A' respectively, and that these two points mark the limits of its vibration. Hence the line AA' is the extent of the particle's vibration. But by drawing parallels to the initial line through A and A' it will be seen, by reference to the definition in § 6, that AA' is also the amplitude of the wave. 'Extent of particle-vibration' and 'amplitude of corresponding wave' are, therefore, only different ways of expressing the same thing.

10.

When a series of continuous equal waves are being transmitted, each particle, after completing one vibration, will instantly commence another precisely equal vibration, and go on doing so as long as the transmission of waves is maintained. This

kind of motion, in which the same movement is continuously repeated in successive equal intervals of time, is called 'periodic,' and the time which any one of the movements occupies is called its 'period.' Thus, to continuous equal waves correspond continuous periodic particle-vibrations.

We will next compare the periods of the vibrations which produce waves of different lengths advancing at the same speed.

In Fig. 8, waves of three different lengths are represented. One wave of (1) is as long as two of

(2), and as three of (3). Therefore, in virtue of the fact proved in § 9, a particle makes one complete