WAVE THEORY 427 parent film is backed by a perfect reflector, no colours should be visible, all the light being ultimately reflected, whatever the wave length may be. The experiment may be tried with a thin layer of gelatin on a polished silver plate. In other cases where a different result is observed, the inference is that either the metal does not reflect perfectly, or else that the material of which the film is composed is not sufficiently transparent. Colours Theory and observation alike show that the transmitted colours f of a thin plate, e.g., a soap film or a layer of air, arc very inferior ..ncient to those reflected. Specimens of ancient glass, which have under pass, gone superficial decomposition, on the other hand, sometimes show transmitted colours of remarkable brilliancy. The probable ex planation, suggested by Brewster, is that we have here to deal not merely with one, but with a series of thin plates of not very different thicknesses. It is evident that with such a series the transmitted colours would be much purer, and the reflected much brighter, than usual. If the thicknesses are strictly equal, certain wave-lengths must still be absolutely missing in the reflected light ; while on the other hand a constancy of the interval between the plates will in general lead to a special preponderance of light of some other wave-length for which all the component parts as they ultimately emerge are in agreement as to phase. 1 Colours All that can be expected from a physical theory is the dctermina- >f New- tion of the composition of the light reflected from or transmitted
- ou s by a thin plate in terms of the composition of the incident light.
- cale. The further question of the chromatic character of the mixtures
thus obtained belongs rather to physiological optics, and cannot be answered without a complete knowledge of the chromatic re lations of the spectral colours themselves. Experiments upon this subject have been made by various observers, and especially by Max well, 2 who has exhibited his results on a colour diagram as used by Newton. A calculation of the colours of thin plates, based upon Maxwell s data, and accompanied by a drawing showing the curve representative of the entire series up to the fifth order, has recently been published ; 3 and to this the reader who desires further infor mation must be referred, with the remark that the true colours are not seen in the usual manner of operating with a plate of air enclosed between glass surfaces, on account of the contamination with white light reflected at the other surfaces of the glasses. This objection is avoided when a soap film is employed, to the manifest advantage of the darker colours, such as the red of the first order. The colours of Newton s scale are met with also in the light transmitted by a somewhat thin plate of doubly-refracting material, such as mica, the plane of analysis being perpendicular to that of primitive polarization. The same series of colours occur also in other optical experi ments, e.g., at the centre of the illuminated area when light issuing from a point passes through a small round aperture in an otherwise opaque screen ( 10). The colours of which we have been speaking are those formed at nearly perpendicular incidence, so that the retardation (reckoned as a distance), viz. , 2/^cosa , is sensibly independent of A. This state of things may be greatly departed from when the thin plate is rarer than its surroundings, and the incidence is such that a is nearly equal to 90, for then, in consequence of the powerful dispersion, cos a may vary greatly as we pass from one colour to another. Under these circumstances the series of colours entirely alters its character, and the bands (corresponding to a graduated thickness) may even lose their coloration, becoming sensibly black and white through many alternations. 4 The general explanation of this remarkable pheno menon was suggested b}- Newton, but it does not appear to have been followed out in accordance with the wave theory. Let us suppose that plane waves of white light travelling in glass are incident at angle a upon a plate of air, which is bounded again on the other side by glass. If /x be the index of the glass, a the angle of refraction, then sino = ^sino; and the retardation, ex pressed by the equivalent distance in air, is 2t sec o - /j. . 2t tan a! sin a = It cos a ; and the retardation in iihasc is 21 cos a /, A being as usual the wave length in air. The first thing to be noticed is that, when a approaches the critical angle, cos a becomes as small as we please, and that conse quently the retardation corresponding to a given thickness is very much less than at perpendicular incidence. Hence the glass surfaces need not be so close as usual. A second feature is the increased brilliancy of the light. Ac cording to (7) the intensity of the reflected light when at a maximum (sinj:5 = l) is 4c/(l + c-)-. At perpendicular incidence e is about J, and the intensity is somewhat small; but, as coso approaches zero, c approaches unity ( 2G), and the brilliancy is much increased. IUUB uiBpiacciueut in piiose us won ns in amplitude, instead ol l fluting merely to ititensitiet. 2 Maxwell, "Theory of Compound Colours." J /iil. Trans., 18CO. 3 Edin. Trans., 1887. 4 Newton s Optics, l>k. ii. ; Fox Tulbot, I hil. Mag., is. p. 401, 16 But the peculiarity which most demands attention is the lessened influence of a variation in A upon the phase-retardation. A diminution of A. of itself increases the retardation of phase, but, since waves of shorter wave-length are more refrangible, this effect may be more or less perfectly compensated by the greater obliquity, and consequent diminution in the value of cos a . "We will investi gate the conditions under which the retardation of phase is stationary in spite of a variation of A. In order that A^cosa may be stationary, we must have A sin a! da! + cos a d = , where (o being constant) cos a da = sin o dfj. . Thus giving a when the relation between ju and A is known. According to Cauchy s formula, which represents the facts very well throughout most of the visible spectrum, M = A + BA- 2 ....... (10), so that eotV-^- 2 -*^ ..... (11). A^ p If we take, as for Chance s "extra-dense flint," B= 984 x 10 " 10 , and, as for the soda lines, /j. = l Q5, A = 5 S9 x 10 " 5 , we get a = 79 30 . At this angle of refraction, and with this kind of glass, the retardation of phase is accordingly nearly independent of wave-length, and therefore the bands formed, as the thickness varies, are approxi mately achromatic. Perfect achromatism would be possible only under a law of dispersion ^ 2 = A -B A 2 . If the source of light be distant and very small, the black bands are wonderfully fine and numerous. The experiment is best made (after Newton) with a right-angled prism, whose hypothenusal surface may be brought into approximate contact with a plate of black glass. The bands should be observed with a convex lens, of about 8 inches focus. If the eye be at twice this distance from the prism, and the lens be held midway between, the advantages are combined of a large field and of maximum distinctness. If Newton s rings are examined through a prism, some very Effect of remarkable phenomena are exhibited, described in his twenty-fourth a prism. observation. 5 "When the two object-glasses are laid upon one another, so as to make the rings of the colours appear, though with my naked eye I could not discern above eight or nine of those rings, yet by viewing them through a prism I could see a far greater multi tude, insomuch that I could number more than forty ..... And I believe that the experiment may be improved to the discovery of far greater numbers ..... But it was on but one side of these rings, namely, that towards which the refraction was made, which by the refraction was rendered distinct, and the other side became more confused than when viewed with the naked eye ..... "I have sometimes so laid one object-glass upon the other that to the naked eye they have all over seemed uniformly white, without the least appearance of any of the coloured rings ; and yet by viewing them through a prism great multitudes of those rings have discovered themselves." Newton was evidently much struck with these "so odd circum stances ; " and he explains the occurrence of the rings at unusual thicknesses as due to the dispersing power of the prism. The blue system being more refracted than the red, it is possible under cer tain conditions that the n th blue ring may be so much displaced relatively to the corresponding red ring as at one part of the cir cumference to compensate for the different diameters. A white stripe may thus be formed in a situation where without the prism the mixture of colours would be complete, so far as could be judged by the eye. The simplest case that can be considered is when the " thin plate " is bounded by plane surfaces inclined to one another at a small angle. By drawing back the prism (whose edge is parallel to the intersection of the above-mentioned planes) it will always be possible so to adjust the effective dispersing power as to bring the ?i th bars to coincidence for any two assigned colours, and therefore approximately for the entire spectrum. The formation of the achromatic band, or rather central black band, depends indeed upon the same principles as the fictitious shifting of the centre of a system of Fresnel s bands when viewed through a prism. But neither Newton nor, as would appear, any of his successors Condi- has explained why the bands should be more numerous than usual, tioii of and under certain conditions sensibly achromatic for a large mini- achro- ber of alternations. It is evident that, in the particular case of the matism. wedge-shaped plate above specified, such a result would not occur. The width of the bands for any colour would be proportional to A, as well after the displacement by the prism as before ; and the succession of colours formed in white light and the number of perceptible bands would be much as usual. _
5 Newton s Optics. See also Place, Pogg. Ann., cxiv. p. 504, 1SG1.