a positive value to the fraction. At the edges of the band, we have

153. Absorption. Absorption has so far not been taken into account though in Art. (151) the coefficient k involves a gradual extinction of light as the wave proceeds. This extinction was made to depend on the imparting of energy from the vibrating electron to the medium. The particle takes up energy from the incident light and communicates it to the æther in the form of a vibration having the same period as the incident light. This is a case of scattering of light and not of absorption. Analytically, of course, we may include absorption if the value of k is increased beyond the amount required for the scattering. This has been very generally done by various writers, but there is no theoretical justification for it.

Absorption is not brought about by anything analogous to ordinary friction, but most probably by sudden changes in the vibratory motion due to molecular impacts. We may take as typical, equation (20) Art. 146, which represents the motion of a vibrating particle subject to a periodic force. The second term quickly subsides if the particle is left to itself, but is constantly being renewed by molecular impacts. The free vibration therefore never disappears and its energy is derived from that of the incident vibration. It is worth while looking a little more closely at this question. Absorption means the gradual decay of the forced vibration into the vibrations and translatory motions corresponding to the temperature of the body, and this takes place in the first instance by a change of the forced vibration into the vibrations of the free periods. Molecular impacts sufficiently account for this first step. Two points should be specially noted. The forced vibration as indicated by the first term in (20) is not disturbed by the impact. Its phase and amplitude remain the same and persist throughout the motion. Secondly, and this is very important—the free vibration does not represent a simple periodic motion. The waves sent out are the less homogeneous the more quickly the free vibrations decay. Molecular impacts therefore increase the possible free periods of motion. If luminous the body would radiate in periods extending over a finite range on both sides of 2π/√n2 – k2, and by the principle of exchange, it must be capable of absorbing those periods which it can radiate.

The periods contained in the second term of (20) include 2π/ which is that of the force, and this has to be taken into account in judging of the effect of the free periods on absorption and wave velocity. From the principle of conservation of energy, we may predict that the effect of that portion of the free vibration which has a period 2π/w will be to reduce the amplitude of the forced vibration, because it is the energy of the forced vibration which by the supposed impact supplies the energy of the rest of the motion. Hence instead of A, as given in (41), we must introduce a value reduced in some unknown ratio. This does not affect our final equations (48) as they contain undetermined factors M1, M2.

Again the principle of conservative energy tells us that the transmitted light must be diminished by molecular impacts, hence the second term of (20) must contain a vibration of period 2π/w which as regards phase is a right angle behind that of the incident light. While B in the simple theory of Art. 149 was zero, we must now conclude that one effect of molecular impact is the introduction of a real positive value for B. That value need not however be that arrived at in Art. 151 by the assumption of a frictional term.

For non-conductors, for which C = 0, equations (39) and (40) become

If ко is determined by observation, the first of these equations gives

v2-1=k2 + NAeV2,

or substituting the more general value of A as determined by (45),

If the absorption band ranges over a finite width between values of n = n1 and n = n2, the absorption being proportional to (n − nı) (n − në), we may substitute the right-hand side of (54) for the second term of (57). If κ, is to be obtained from the general theory, both equations (56) must be used. Eliminating Eliminating between them, we find for the general equation of v

If we substitute terms of the form (52) instead of (45) for A, we take account of the loss of energy by radiation, and have then obtained the most general theory of refraction that can at present be formed. The quantity B is not known, but we shall probably not go far wrong in taking it to be proportional to A.

154. Selective refraction. The phenomena which have called forth the theoretical discussions of this Chapter have been grouped together under the name Anomalous Dispersion." But we are now prepared to say that there is nothing anomalous in the effect of absorption on refraction, and that the ordinary or "normal" dispersion is only a particular case of the "anomalous one. Under these circumstances the name is misleading, and I therefore introduce the more appropriate one of "Selective Refraction" and "Selective Dispersion."

The experimental illustration of selective refraction is rendered somewhat difficult by the fact that the substances which show the effects are all highly absorbent. With a hollow prism filled with a strong solution of fuchsin or cyanin, it may easily be demonstrated that the red of the spectrum is more refracted than the violet, but dispersion in the immediate neighbourhood of the absorption band is too great to make exact measurements in that region possible. Kundt originated a method of observation which is often employed. The vertical slit of a spectroscope is illuminated by projecting upon it the image of a horizontal slit, through which white light is passed. If the horizontal slit be narrow, an almost linear spectrum is seen, running along a horizontal line. The position of this horizontal line may be marked. If now a prism filled with a substance showing selective refraction be interposed between the horizontal slit and its image, the refracting edge of this prism being horizontal and downwards, the line of the observed spectrum will no longer be straight. Were the prism filled with water, the spectrum would run upwards in a curved line from red to violet. A curve running downwards from red to violet would indicate a refractive index diminishing with increasing frequencies. Refractive indices smaller than one, showing a velocity of light greater than that of empty space, would be indicated by displacement below that of the original linear spectrum. For purposes of illustration, and for measurements when the angle of the prism is small, this method is very successful.

The simplest case of selective refraction is shown by sodium vapour, as the absorption is here confined to two regions, each of which covers only a very narrow range of wave-lengths. In other words, the refraction is affected by absorption "lines as distinguished from absorption bands. The selective refraction of a luminous conical sodium flame was first shown by Kundt, who however did not investigate the specially interesting region which lies between the absorption lines. This has been done by Becquerel. Plate II. Fig. 6 is a reproduction of one of Mr Becquerel's photographs, the red end being to the right. The sodium vapour was used in the form of a luminous flame formed by a special device into a prismatic shape.

The horizontal black line marks the horizontal linear spectrum in its original position. The horizontal portion of the white band, the centre of which is slightly raised above the black line, shows that at a short distance from the double sodium line there is a slight displacement upwards indicating a refractive index somewhat greater than one. The nearly vertical branches of the curve indicate a considerable dispersion close to and between the absorption lines. The course of this dispersion is better studied in Fig. 168, which has been drawn

from the measurements given by Mr Henri Becquerel*, upward displacements being approximately proportional to μ-1. It will be seen that in accordance with the previous theory, the refractive index rapidly increases as we approach each absorption line from the red end, and that the light which vibrates just a little more quickly than that corresponding to the absorption band has its velocity increased. Mr Becquerel calculates approximately that the light in close proximity to D2 and on its violet side has

2

a refractive index of 9988, so that its velocity is about 1% greater

than that in empty space. Concentrated solutions of colouring matters exhibit the phenomena of selective refraction, but here the theory is complicated by the fact that the absorption extends over a wide range of wave-lengths. Some of these colouring matters may be shaped into solid prisms of small angle, by means of which the refractive indices for different periods and the coefficients of absorption may be measured. Pflügert takes a few drops of a concentrated solution of the colouring matters in alcohol, and runs the solution into the two wedge-shaped spaces between a glass plate and a wide glass tube. As the solvent evaporates, it leaves behind a double prism. Amongst many prisms made in this fashion, a few may be found with surfaces sufficiently good to render optical investigation of refractive indices possible. The prisms used by Pflüger had a refracting angle of from 40-130 seconds of arc, and the refractive indices could be determined throughout the absorption band. It is a special merit of Pflüger's

* Comptes Rendus cxxvIII. p. 145 (1899).

+ Wied. Ann. LVI. p. 412 (1895) and LXV. p. 173 (1898).

investigations that he determined also the coefficients of absorption for the different wave-lengths. As a thickness of very few wave-lengths is sufficient to extinguish the light, the plates used for the purpose had a thickness of less than half a wave-length. In Figure 169

Pflüger's curves for cyanin are reproduced, the curves of refractive index and coefficient of extinction being marked B and A respectively. I have added in dotted lines an assumed absorption curve following the law discussed in Art. 150, the constants being roughly adjusted so as to fit the edges of the absorption band. A second dotted line shows the curve of refraction (B') calculated from equation (55) after substitution of v2 - k2 for μ2. The value of ẞ was determined so as to give roughly the proper quantity for the difference in the refractive indices near the two edges of the absorption band, and a constant term has been added to represent the effect of infra-red and ultra-violet absorption. The correspondence between calculated and experimental values might be made closer if instead of a constant term, one varying with the wave-length had been taken, but in view of the fact that the assumed law of absorption only approximately represents the facts, it seems unnecessary to seek for a closer agreement of the refraction curves. The more sudden fall and rise of the calculated dispersion curve near the green boundary of the absorption band is due to the fact that the actual absorption curve does not show the rapid increase of absorption indicated by the assumed curve.

155. Metallic Reflexion. We include under the term "metallic" reflexion, all cases in which the greater portion of the incident light is returned, in consequence, as it will appear, of the absorptive power of the medium. If the amplitudes of the incident, reflected and