to move the ball from its original position in opposition to the elasticity of the wire increases in the same ratio as the amount of displacement effected.

Let the weights be now removed and when the ball has returned to its original position, let it be pressed down with the fingers about two centimeters; then inasmuch as it is kept in this position, the pressure downwards exerted must be identical with the weight of 100 grammes, which was before necessary to effect this elongation, and when the ball is set free it returns with this force to its position of equilibrium.

When, however, it has reached the position of equilibrium it does not at once come to rest, but continues to perform upward and downward movements which are slow enough to permit them to be counted. If the ball be now depressed to the extent of 4 centimeters, and then be set at liberty, it has twice as far to go from its extreme point to the position of equilibrium, or the extent (or amplitude) of its vibration is now doubled. If its vibrations are now counted the same number of vibrations will be found as in the former case, for since not only the space traversed but also the expression of force of the tense spiral wire has now been doubled, the greater space must be traversed in the same time. Nor is any alteration observable in the number of vibrations when the ball is drawn down to the extent of 6 centimeters from its position of rest, although the amplitude of its vibration is increased threefold.

From this it appears that the number of vibrations is dependent exclusively upon the nature of the vibrating body—upon its internal forces, if we may so speak, --but in no way upon the amount of the external force applied to it; the amount of force applied to it finds its

expression in the amplitude of the vibration. When the ball is depressed four centimeters, the hand has not only to exercise twice as much force, but it has to traverse twice the distance that it has when it is only depressed two centimeters. The work which must be performed to overcome the elastic force of the wire in the former case is therefore four times as great as in the latter, and if with three times the force the ball be moved over three times the space, nine times the amount of force used in the first instance has to be applied. When the hand is removed the work performed by it is transferred to the ball, and expresses itself in the energy of its vibrating movements. By virtue of this energy the ball, until it comes to rest, performs the same amount of work which was applied to it to set it in movement.

From these considerations it results that the energy of the vibrating movement is proportional to the square of the amplitude of the vibrations.

The facts taught by the vibrating ball are applicable alike to the vibrations of sound and of light. The tint of colour is dependent on the frequency; the intensity (or energy) of light on the liveliness of the vibrations. Whilst the former depend on the number of the vibrations, the latter are measured by the square of the amplitude of the vibrations.

CHAPTER XVII.

HUYGHENS' PRINCIPLE.

97. LIGHT consists of a very minute vibrating movement of an elastic medium, which is propagated with great rapidity, but not instantaneously, in straight lines that proceed like the radii of a sphere from a central point common to all.'

Hooke, the accomplished friend and countryman of Newton, who wrote the date 1664 under these words, may be regarded as the first who clearly seized and expressed the fundamental idea of the doctrine of luminous waves. Nevertheless he did not advance so far as to explain the refraction of light by undulatory movement; and he failed because this fundamental idea, in order to be applicable to all the phenomena of light, required still a very important addition to complete and perfect it. It was reserved for Hooke's genial contemporary, Huyghens,† to fill this hiatus, and to become the real founder of the undulatory theory of light.

The theory of Huyghens, so named to do honour to its discoverer, is in fact the egg of Columbus, a simple solution of many complex and enigmatical phenomena, and whilst an attempt is here made to render it intel

*Micrographia, Observat. ix.

Tractatus de Lumine, 1690.

A

FIG. 137.

B

·ligible, no very great strain will be exerted on ordinary powers of imagination. When an undulatory movement propagates itself through an elastic medium, every particle imitates the movement of the particle B' first excited. But every particle stands in regard to the adjoining ones in exactly the same relation that the first particle did to its neighbours, and consequently must exert upon those that surround it exactly the same influence as the first. Every vibrating particle is therefore to be regarded as if it were the originally excited particle of a wave system; and as the innumerable and simultaneous elementary' wave systems co-operate with one another at each instant in accordance with the principle of interference, we obtain exactly that principal wave system' by which the elastic medium appears at any moment to be moved.

Huyghens' principle.

If, for example, all points of the circular or spherical wave BC (fig. 137) which take origin from the centre of disturbance A be regarded as new centres of disturbance, after a little while an innumerable series of elementary waves of equal size will have formed around them, which are represented in the figure by small arcs. The circle B'C' described around the centre A, which all the elementary waves touch at their most distant point, represents the extreme limits to which the undulatory movement has in the meanwhile been propagated. The state of oscillation which previously affected the wave BC is now transferred to the

circle B'C', to which all elementary waves reach with equal conditions of oscillation. The wave BC has thus propagated itself by means of the elementary waves in the same form and with the same rapidity to B′ C", as if it proceeded directly from the original point of disturbance A.

The same result is thus obtained whether we admit a direct propagation of a single wave centre outwards, or an indirect propagation effected by innumerable elementary waves. Nevertheless the two modes of explanation are essentially different, and the latter is alone true to nature, for it alone gives the requisite consideration to the various relations that occur between the particles of an elastic medium. The former more simple mode of explanation may, however, be admitted if, as in the preceding Chapter, we are dealing with those characters of wave movement which are common to both methods of propagation. As long as a wave movement is propagated without disturbance, the elementary waves withdraw themselves from observation because they proceed by their co-operation to produce the chief waves. They immediately appear independently, however, if their adjoining waves are in any way suppressed. If, for example (in fig. 137), the wave BC proceeding from A passes through the opening BC of a screen, it continues its course undisturbed between the two marginal rays AB and AC, whilst the elementary waves proceeding from their points between B and C combine in the manner above described to form the chief wave B′ C. The elementary waves B' b and Cc proceeding from the marginal points B and C remain partially isolated, and transfer a movement which, in comparison with the main wave, is, as may be