A single pulsation of air is audible as a noise, though it does not convey the idea of pitch ; but even in the most transitory light, such as that of the electric spark, the phenomena of dispersion and inter ference indicate that we have to deal with a succession of a great num ber of similar undulations. Returning, for simplicity of conception, to the illustration of the rope, let us therefore suppose that the operator moves his hand backwards and forwards in a regular periodic, manner. The rope will be thrown into the form of a sinuous curve, which travels along it The distance from any particle to the next before or behind which is in the same state of motion, is called the length of a ware. It is evident that a single wave-length comprises two bow.shaped portions of the curve, the displacements in which are on opposite eides of the mean position,and which may be distinguialied as positive and negative; and also that any two points distant by half a wave's length are in exactly opposite states of displacement and motion, at least if we sup pose the positive and negative portions of the curve exactly alike. If we reflect on the motion of any particular particle, we shall readily see that it goes through its changes once while the waveform progresses by one wave's length. Hence, if r be the velocity of propagation, A the length of a wave, and T the periodic time of vibration of a single particle, we have the fundamental relation A = which applies to undulations in general, since what was said in the ease of the rope holds good generally. In the ease of light, the absence of prismatic colours when a star is displaced by aberration from its mean position, the AMOCO of changes of colour when one of Jupiter's satellites enters or emerges from his shadow, and the existence of periodic stars, such AS Algol, which rapidly change in brightness without changing in colour, indicate that the velocity of propagation of light in vacuo is the same for all colours. The phenomena of interference furnish us with means of measuring the length of a wave of light, which is found to vary from about 266 to 167 ten-millionths of an inch, in passing from the extreme red to the extreme violet. We infer, therefore, from the known velocity of propagation, that from about 458 to 727 millions of millions of vibrations must take place in one second.
The locus of the particles, which at a given instant are in the same state or phase of their motion, is called the front of a rare. When light di rerges in the first instance in vacuum, or any singly refracting medium, from an element of a self-luminous body, the front of a wave is of course spherical, or, at a sufficient distance from the body, when we have to consider a small portion only of the front, sensibly plane, but after reflection or refraction it may be of any other form.
We now come to a principle of constant application in the theory of undulations, which is called Ilnygens'e principle. It may be thus stated :—The front of a wave of light, either at a given instant, or as the parts of it arrive in succession at a given surface, may be divided into elements, of which each is conceived to be the centre of an ele mentary disturbance which spreads in all directions with the velocity appropriate to the medium in which it is propagated. The disturbance in front of the primary wave may be regarded as the aggregate of the elementary disturbances due to these secondary waves; and as these are insensible when taken separately, the aggregate disturbance will not–be sensible except in the immediate neighbourhood of the enve lope, or surface of ultimate intersection, of the secondary waves. If
we confine our attention to a small portion of the primary wave, we shall have a corresponding small portion of the envelope, or general wave, fn its advanced position, in the neighbourhood of which alone the disturbance due to the small element of the first wave is sensible. Hence the conrse of a ray from any point of the first wave is the line along which the point of ultimate intersection of the secondary waves which start from the neighbourhood of that point of the first wave is displaced. This is evidently the straight line joining the point last mentioned with the point of ultimate intersection at a given instant ; or, again. the straight line joining the point in question of the first wave with the point in which the secondary wave thence diverging is touched by the general envelope.
The legitimacy of thus conceiving a wave to be broken np is a direct consequence of the general dynamical principle of the co-existenco of small motions. The secondary waves will have an envelope behind as well as in front of the primary wave; yet we must not infer that the latter is propagated backwards as well as forwards. The explanation of the non-propagation in a backward direction, notwithstanding the legitimacy of thus supposing a wave broken up, belongs to the dyna mical theory of diffraction, and is much too difficult a subject to be entered upon here. (See a paper in the Cambridge Philosophical Transactions,' voL ix., p. 1.) The complete explanation of the existence of rays requires this principle to be combined with the principle of interference, bnt for the present we shall confine ourselves to its application to the demon stration, according to the undulatory theory, of the laws of reflection and refraction. We may notice, however, in the mean time some features of the propagation of light in a uniform medium.
Conceive, then, a surface of any form to he at a given instant the front of a wave propagated in a uniform medium, and first suppose the velocity of propagation the same in all directiona. If we conceive the wave broken up as above explained, the secondary waves, which will be all of the Brune magnitude, will by our supposition be spherical; and by very simple geometry we arrive at the following construction for determining the form of the wave in its onward course. At all points of the front of the wave at the time t draw normals, at the side towards which the wave is travelling, equal in length to v ; the locus of the extremities of these normal' will be the front of the wave at the time t + e, and the normals themselves will be the courses of the rays. We learn also that it is not any arbitrarily chosen system of straight lines, the equations of which contain two, arbitrary parameters, that can represent a possible system of rays ; the lines meat admit of being cut orthogonally by a system of surfaces. This geometrical property, so readily suggested by the theory of undulations, may of course be demonstrated as a consequence of the geometrical laws of reflection and refraction independently of any theory, the system being supposed to consist of rays which, having originally emanated from a point, have undergone any number of reflections and refractions. If the medium in which the wave is propagated bo uniform, but differently consti tuted in diffirent directions, as in the case of Iceland spar, for example, the velocity of propagation will vary with the direction, the secondary waves will no longer be spherical, and the course of a ray, while still rectilinear, will no longer be perpendicular to the front of the wave.