Supposing, therefore, for the sake of simplicity, that the light is ucideut perpendicularly, restricting ourselves to the two most im portant pencils, and considering first the reflected light, we see that he light reflected from the under surface of the thin plate of air has had to travel a distance 2n in air more than the other stream, n being the distance between the lenses at the point under consideration, or the thickness of the plate of air. We might, therefore, perhaps, aspect that the vibrations in the two streams would be in perfect eccordauce when n was equal to zero, or a multiple of AA, and in apposition when n was an odd multiple of This would give cor rectly the law of the variation of the radii of the rings, since D varies AS the square of the radius drawn from the point of contact of the lenses, with the single but important exception, that the places of the bright and dark rings are interchanged. But we must remark, that the two reflections take place under opposite circumstances, one at the surface of a rarer, the other at the surface of a denser medium. Varioua dynamical analogies, such as the reflection of sound from the end of a tube, according as it is closed or open, the reflection of sound at the common surface of two gasca which are supposed not to mix, would make it more probable than the contrary supposition that in one of these two reflections there should be a change of sign, in the other not. A change of sign is equivalent to a change in the length of the path of one of the streams amounting to half an undulation. This change being admitted, theory assigns correctly the law of the variation of the radii of the bright and dark rings. And not only the law of variation, but the absolute of the rings may be assigned d priori, since the length of a wave of light is known by other observa tions ; and the magnitude so assigned is in conformity with experiment.
The explanation of the variation of the scale of the rings with the colour, and of the obliteration of the rings beyond the seventh or thereabouts by overlapping, when the incident light is white, is the sane as in other cases of interference. Moreover, if a liquid such as water be interposed between the lenses in place of air, since light travels more slowly in water than in air, in the proportion of 1 to p., the same kind of ring which with air is found at a spot where the distance between the lenses is n, is found with water where it is only which explains the law of variation of the radii of the rings with the refractive index of the interposed medium.
The explanation of the transmitted rings is perfectly similar, the chief differences being, first, that as the interfering stream have been refracted alike, and one of them iu addition twice reflected, there is no change of sign, or the interference is determined simply by the difference of path, without the addition of the half undulation ; and secondly, that the interfering streams are very unequal in intensity, and therefore with homogeneous light the minima are very far from being absolutely black. Thus when light is incident perpendicularly, whether externally or Internally, at the common surface of crown glass and air, only about the tai part of the incident light is reflected, the remaining 'Wile being transmitted. Hence by two such reflections the intensity is reduced in the proportion of 625 to 1; and if we represent by unity the intensity of the light transmitted across the thin plate without reflection, the intensity of the twice reflected light must be represented by The question may naturally be asked, How can such feeble light by interfering with the former give rise to any sensible tinge I The explanation of this paradox is derived from the consideration that in interference we must compound not inten sities but vibrations, and thence deduce the intensity by taking the square of the coefficient of vibration. Thus, taking the above numbers
as correct, we learn that the coefficient of vibration will be reduced by one reflection in the proportion of 5 to 1 only, and by two in the proportion of 25 to 1. Hence if we take the coefficient of vibration in the simply transmitted stream as unity, that in the twice reflected stream will be and therefore that in the resultant stream will vary between the limits 1 ± h, and the intensity will vary between the limits (1 ± h)', or 1 yearly ; so that the difference between the limits is as much as or nearly ,nth of the mean intensity.
Next, suppose the light incident obliquely. If ft be the angle of incidence on the first surface of the upper lens, or which is the same (the lens for this purpose being treated as a plate bounded by parallel surfaces), the angle of refraction into the thin plate of air, it may be shown, as in Airy's ' Tracts' (' Undulatory Theory,' art. 64), or still more simply by referring everything to sections of the fronts of the waves by the plane of incidence instead of to rays, that the retardation due to the double transit across the plate is 2 D cos t3, in place of 2 n, as at a perpendicular incidence. This explains the law of the variation of the rings with the obliquity.
From the explanation hitherto given it might seem that in the case of the reflected rings the minima, though nearly, ought not to be perfectly black. For the stream reflected at the second surface of the, thin plate has to undergo two more refractions than that reflected at the first surface, and at each refraction a small portion of the incident light would be lost to it by reflection. Freenel first showed (‘ Annalea de Chimie,' tom. 23, p. 129), by taking account of the infinite number of reflections, as Poisson had previously done in the case of a perpen dicular incidence, that the minima in the reflected rings ought at any incidence to be perfectly black, and that, without assuming anything relative to the law of intensity in reflection beyond a law discovered experimentally by Arago, that at any obliquity light is reflected in the same proportion at the first and second surfaces of a transparent plate. For a very simple demonstration at the same time of Arago's law, and of the loss of a half undulation, on the assumption merely that the forces acting depend only on the positions of the particles, the reader is referred to a paper in the Cambridge and Dublin Mathe matical Journal' (vol. iv. p. 1). A complete investigation of the intensities of the reflected and transmitted rings, in which account is taken of the infinite number of reflections, will be found in Airy's ' Tract,' arts. 64-67. The result is where a denotes the coefficient of vibration for the incident light, v the retardation, 2 n cost), and 1 : e the ratio in which the coefficient of vibration is altered by one reflection. The sum of the two expres sions is always equal to at, which shows that the reflected and refracted systems are complementary to each other, conformably to observation.