More accurate measurements show that the velocity of light in vacuo is about 299,860,000 metres per second. The velocity of light in vacuo, and in media which do not produce dispersion, is practically the same for all colours, and appears from theoretical considerations to depend upon the properties of the medium in which the light is being propagated. Both theory and experiment agree in showing that the colour of light depends upon the period of vibration; or, since the velocity is approximately independent of the colour, the colour may also be considered to depend on the wave-length. The periods of the red waves are the longest, whilst those of the violet are the shortest; accordingly the wave-lengths of the red waves are longer than those of the violet waves. The imagination may be assisted by considering the analogous problem of waves of sound; for it is known that the velocity of different notes is very approximately the same, and that the pitch of a note is determined either by its period or its wave-length, notes of low pitch corresponding to waves of long period or large wave-length. There is however an important distinction between sound and light; for if the period of one note is double that of another, the notes stand to one another in the relation of octaves. Nothing corresponds to this in the case of light, and one reason of this is, that the sensitiveness of the ear extends over several octaves, whereas the sensitiveness of the eye is limited to less than an octave. A table of the wave-lengths of the principal lines of the spectrum will hereafter be given; but it will be well to mention, that the wave-lengths, in tenth-metres, of the extreme red and violet rays are about 7604 and 3933 respectively. A tenth-metre is 10-10 of a metre. 10. The intensity of light has usually been considered to be proportional to the square of the amplitude. This may be seen as follows. Let O be a source of light, and let ds, dS' be elements of the surfaces of two spheres, whose common centre is 0, which are cut off by a cone whose vertex is 0. Let r, r' be the radii of the spheres, I, I' the intensities at dS, ds. Since the total quantity of light which falls on the two elements is equal, Now if A, A' be the amplitudes of the spherical waves at dS, ds', every dynamical theory of light shows, that A and A' are inversely proportional to r and r'; whence I| A2 = I'/A'3, from which we infer that the intensity is proportional to the square of the amplitude. The undulatory theory supposes that a state of vibration is propagated through the ether; hence every element of volume possesses energy, which travels through space with the velocity of the wave. Modern writers have thus been led to the conclusion that an intimate connection exists between energy and intensity, but as regards the mathematical form of the connection between the two, opinions are not altogether uniform. Lord Rayleigh measures the intensity of light, by the rate at which energy is propagated across a given area parallel to the waves1; on the other hand, writers on the electromagnetic theory measure the intensity, by the average energy per unit of volume. Now so long as we are considering the propagation of light in a single isotropic medium, it is not of much consequence which definition we adopt, since the ratio of the intensities of two lights is proportional to the ratio of the squares of their amplitudes; but when we are considering the refraction of light, the ratio of the intensities of the incident and the refracted light will not be proportional to the squares of their amplitudes, but will contain a factor, whose value will depend upon which definition we adopt. It will be advantageous to adopt a definition, which will make the ratio of the intensities of the incident to the refracted light the same in the various dynamical theories which we shall hereafter consider, and I shall therefore define the intensity of light to be measured by the average energy per unit of volume. It is well known, that when the medium is a gas or an elastic solid, the total energy due to wave motion is half kinetic and half potential; it can also be shown in the case of electromagnetic waves, that the total energy is half electrostatic and half electrokinetic. If therefore we assume that a similar proposition is true in the case of the ether, with regard to whose physical properties we have not at present made any 1 "Wave Theory," Encyclopædia Britannica. DIRECTION OF VIBRATION. 9 hypothesis, it follows from (2) that the total energy E per unit We therefore see that E consists of a non-periodic and a periodic term, the latter of which fluctuates in value. The former term represents what we have called the average energy per unit of volume, and accordingly the intensity is measured by the quantity 2π2A2p/T2, and is directly proportional to the product of the density and the square of the amplitude, and inversely proportional to the square of the period. 11. Up to the present time we have been considering the propagation of waves in an elastic medium, and have not made any supposition as to the direction of vibration. We now come to a point of capital importance, which constitutes one of the fundamental distinctions between waves of sound and waves of light. It is well known that in a plane wave of sound, the displacement is perpendicular to the wave front; but in the case of plane waves of light, the displacement lies in the front of the wave. This is established experimentally in the following manner. There are certain crystals, of which Iceland spar is a good example, which possess the power of dividing a given incident ray into two refracted rays, and from this property such substances are called doubly refracting crystals. Iceland spar is called a uniaxal crystal, owing to the fact that there is a certain direction, called the optic axis of the crystal, with respect to which the properties of the crystal are symmetrical. The symmetry of uniaxal crystals is therefore of the same kind as that of a solid of revolution. There is a second class of doubly refracting crystals, called biaxal crystals, which possess two optic axes, with respect to neither of which the properties of the crystal are symmetrical. Now if either of the two refracted rays, which are produced by a single ray incident upon a plate of Iceland spar, be transmitted through a second plate cut parallel to the axis, it will be found that although this ray is usually divided into two refracted rays, there are four positions, at right angles to one another, in which one or other of the two refracted rays is absent. If the second plate be placed in one of these four positions, and then be turned round an axis perpendicular to its faces, the absent ray immediately appears, and its intensity increases, whilst that of the original ray diminishes; when the plate has been turned through a right angle, the intensity of the ray which was absent is a maximum, whilst the original ray has altogether disappeared; and when the second plate has been turned through two right angles, the original state of things is restored. Now if the vibrations were perpendicular to the wave front, and therefore in an isotropic medium parallel to the ray, the properties of a ray would be the same on every side of it; if however the vibrations were parallel to the wave front, and therefore perpendicular to the ray, we should anticipate that the properties of a ray would be different on different sides of it. We are thus led to the conclusion that the vibrations of light are parallel to the front of the wave, and this conclusion is amply justified by theory and experiment. 12. Let us now suppose that the axis of x is the direction of propagation, and that the axis of y is parallel to the direction of vibration; then if v be the displacement, it follows from (1) that Equation (4) represents what is called a wave of plane polarized light, and the plane of xz, which is the plane to which the vibrations are perpendicular, is called the plane of polarization. It was for many years a disputed point, whether the vibrations of polarized light were in or perpendicular to the plane of polarization, but modern investigations have shown that the latter supposition is the true one. We shall discuss this point in detail in subsequent chapters. We shall now show how two trains of waves, whose wavelengths are equal, and whose planes of polarization are the same, may be compounded. Let the first wave be given by (4), and let the second wave be then if 2π v' = A' cos (x − Vt - e'); $ = (2π/λ) (x − Vt − e), d = (2π/λ) (e − e'); We therefore see that the two waves compound into a single wave, whose amplitude and phase are different from those of either of the original waves. = The amplitude is proportional to (A2 + A22 + 2AA' cos 8)3, and may therefore vary from A- A' to A+ A'; the first of which corresponds to 8 (2n+1) π, or e=e' + (n + 1) λ, and the second to d=2nπ or e=e+nλ. We therefore see, that when the phases differ by an odd multiple of half a wave-length, the intensity of the light due to the superposition of the two waves is diminished; whilst when the difference of phase is an even multiple of half a wave-length, the intensity of the resultant light is increased. = If A A', we see that the intensity of the resultant light vanishes in the former case. Thus the superposition of two lights can produce darkness. This remark will be found to be of great importance in subsequent chapters. 13. We shall now consider the effect of compounding two waves which are polarized in perpendicular planes. in which the first represents a wave polarized in the plane az, and the second a wave polarized in the plane xy. In the notation of the last section, these equations may be written From equation (7), we see that the elements of ether describe ellipses, whose planes are at right angles to the direction of |