VALUE OF THE INDEX OF REFRACTION. 325 359. Since the absorbing power of transparent media is small, the value of the index of refraction will scarcely be affected thereby; we may therefore as a sufficient approximation put q = 0, and (18) becomes whence du2/dr2 is negative; it therefore follows that μ2 decreases as T2 increases. = T Since ρ is the density of ether when loaded with matter, it follows that p>P.; hence when т=0, μ> 1. As T increases, μ2 μ diminishes to unity; it then becomes less than unity, until T attains a value T3, which makes μ=0. When 7 >73, μ2 is negative; and consequently at this point an absorption band commences, which continues until TT2, where T2 is the value of T which makes the denominator vanish. When T = T2, μ2 = ∞; and when TT2, μ2 is a very large positive quantity, and regular refraction begins again. As 7 still further increases, μ2 continues to diminish, until attains a value T1, such that μ2 is again zero; when т> T1, μ2 becomes negative, and remains so for all greater values of T. The medium is therefore absolutely opaque to waves whose periods are greater than 7; it is transparent for waves whose periods are less than 7, and greater than 72; it is opaque for waves whose periods lie between 7, and T., and is transparent for waves of shorter period. T2 If we now suppose that T, corresponds to the double sodium line D, whilst T, corresponds to the hydrogen line F, we shall obtain a mechanical representation of a medium, which has an absorption band in the green; also the dispersion is anomalous, since the value of μ2 when 7 is a little greater than 72, is greater than it is when is a little less than 7. A medium of this kind accordingly represents a substance such as fuchsine, which has an absorption band in the green, and produces anomalous dispersion. T 360. To explain ordinary dispersion, we shall suppose that T is greater than ; then if we put b2 = x2/(1 + Qx2), With the exception of the term involving 72, this value of μ2 is of the same form as Cauchy's formula μ2 = a + c/x2+d/λ' + .... Ketteler1 has however shown that the term Pb22/2 is required to explain the dispersion produced by certain substances. 361. When there are several absorption bands, a molecule of a more complicated character is required; and it has been suggested by Von Helmholtz, that a theory might be constructed by a hypothesis, which practically amounts to assuming that W, and W, consist of a series of terms of the form 3 362. We must now consider the reflection of light at the surface of a medium, which produces anomalous dispersion. 2 In forming the equations of motion by means of the Principle of Least Action, we observe that there are no surface integral terms, which arise from W, and W.; it therefore follows, that the boundary conditions at the common surface of two different media, are unaffected by the presence of the terms depending on the action of the matter. These conditions will therefore be, continuity of the displacements and stresses arising from the action of the ether. With regard to the physical properties of the ether, I shall provisionally adopt the hypothesis of Lord Kelvin, that the latter is to be treated as an elastic medium, whose resistance to compression is a negative quantity, the numerical value of which is slightly less than 3rds of the rigidity. Under these circumstances, the intensities of the reflected and refracted light will be given by Fresnel's formulæ, and so long as μ > 1, the reflection takes place in the same manner as from glass. If however the incident light is white, and lies within the visible spectrum, say between D and F, it follows that for certain rays of the spectrum μ<1, in which case there will be a critical angle. 1 Treatise on Theoretische Optik. SELECTIVE REFLECTION. 327 Hence those rays, for which < sini, will be totally reflected with a change of phase. There will also be another set of rays, for which μ2 is a negative quantity, and we shall now show that these rays are also totally reflected with a change of phase. which shows that the reflection is total, and that there is a change of phase, whose value is determined by (21). Similarly when the light is polarized perpendicularly to the plane of incidence, B' B = = tan (i − r) which shows that the reflection is total, and is accompanied by a change of phase which is determined by equation (22). Since the change of phase is different, according as the light is polarized in or perpendicularly to the plane of incidence, it follows that the reflected light will be elliptically polarized. 364. We must now enquire, how far these results will explain the selective reflection of substances, which produce anomalous dispersion. T3 = T Let 7, be the least value of 7 for which μ= 0, and let 7, be the value for which μo. Then if we suppose that 7, and T2 respectively correspond to the lines D and F, it follows that there will be an absorption band in the green; accordingly the substance will transmit the blue and red rays, and some of the yellow, hence the colour of the transmitted light will be red or reddish blue. Since μ is a real quantity for these rays, the latter will be reflected in the ordinary manner; but the green rays, for which μ is imaginary, will be totally reflected with a change of phase; hence the colour of the reflected light will be green, and the light will be elliptically polarized. In fuchsine the order of colours going up the spectrum is F, G, H, then there is an absorption band, and then come the lines A, B C, D. The absorption is very strong between D and F; accordingly fuchsine is opaque to the green portion of the spectrum, and ought to reflect green light strongly. This agrees with observation. Now although fuchsine is not absolutely opaque to blue and yellow light, the prevailing colour of the transmitted light is rose; accordingly this substance must possess a tendency to transmit red light to a far greater extent than blue or yellow, and consequently ought to reflect the latter colours more strongly than the former one. This is also borne out by observation, since the reflected light is green. 365. Let us now suppose, that the incident light is common light; and that the reflected light is examined through a Nicol's prism, whose principal section is parallel to the plane of incidence. Then the vibrations perpendicular to the plane of incidence cannot get through the Nicol, and may be therefore left out of account. Now the reflected light consists of three portions; (i) the portion for which μ2 is negative, and which begins a little above D and ends a little below F; (ii) a certain portion for which μ2 is positive, and less than unity, and which lies in the neighbourhood of F; (iii) the portion for which μ2 >1, which is regularly reflected, and constitutes the remaining portion of the spectrum. The first portion is by far the most intense, since the reflection is total; the third portion is the least intense; whilst of the second, for which μ<1, those rays for which the critical angle is less than the angle of incidence, will be totally reflected, and those for which it is greater, will be regularly reflected. Now if the angle of incidence is nearly equal to the polarizing angle, it follows that the yellow portion of the incident light, most of which is regularly reflected, will be polarized in the plane of incidence by reflection, and will therefore be unable to get through the Nicol; but the green, and also a portion of the blue in the neighbourhood of F, will get through. The colour of the light, when viewed through a Nicol, will accordingly change from a green to a greenish blue, owing to the absence of the yellow light. CHAPTER XVIII. METALLIC REFLECTION. 366. THE leading experimental facts connected with metallic reflection may be classified as follows. (i) Metals are exceedingly opaque to light, but at the same time reflect a very large proportion of the incident light. (ii) When plane polarized light is incident upon a polished metallic surface, the reflected light is always elliptically polarized, unless the incident light is polarized in or perpendicularly to the plane of incidence, in which case the reflected light is plane polarized. (iii) Metals do not possess a polarizing angle, but there is a certain angle of incidence, for which the intensity of light polarized perpendicularly to the plane of incidence is a minimum. (iv) When the incident light is circularly polarized, there is a certain angle of incidence, for which the reflected light is plane polarized. Whatever the character of the incident light may be, it can always be resolved into two components, which are respectively in and perpendicular to the plane of incidence; and the above experimental results show, that metallic reflection produces a change of phase in one or both of these components. 367. The angle of incidence, for which circularly polarized light is converted into plane polarized light, is called the principal incidence; and the azimuth of the plane of polarization of the reflected light, is called the principal azimuth. The principal azimuth is usually measured from the plane of incidence |