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Rh are given by Lord Rayleigh in a paper “On Achromatic Interference Bands” (Phil. Mag., 1889, 28, pp. 77, 189); see also E. Mascart, Traité d’optique.

In Newton’s rings the variable element is the thickness of the plate, to which the retardation is directly proportional, and in the ideal case the angle of incidence is constant. To observe them the eye is focused upon the thin plate itself, and if the plate is very thin no particular precautions are necessary. As the plate thickens and the order of interference increases, there is more and more demand for homogeneity in the light, and we may have recourse to a sodium-flame or a helium vacuum tube. At the same time the disturbing influence of obliquity increases. Unless the aperture of the eye is reduced, the rays reaching it from even the same point of the plate are differently affected, and complications ensue tending to impair the distinctness of the bands. To obviate this disturbance it is best to work at incidences as nearly as possible perpendicular.

The bands seen when light from a soda flame falls upon nearly parallel surfaces are often employed as a test of flatness. Two flat surfaces can be made to fit, and then the bands are few and broad, if not entirely absent; and, however the surfaces may be presented to one another, the bands should be straight, parallel and equidistant. If this condition be violated, one or other of the surfaces deviates from flatness. In fig. 4, A and B represent the glasses to be tested, and C is a lens of 2 or 3 ft. focal length. Rays diverging from a soda flame at E are rendered parallel by the lens, and after reflection from the surfaces are recombined by the lens at E. To make an observation, the coincidence of the radiant point and its image must be somewhat disturbed, the one being displaced to a position a little beyond, and the other to a position a little in front of the diagram. The eye, protected from the flame by a suitable screen, is placed at the image, and being focused upon AB, sees the field traversed by bands. The reflector D is introduced as a matter of convenience to make the line of vision horizontal.

These bands may be photographed. The lens of the camera takes the place of the eye, and should be as close to the flame as possible. With suitable plates, sensitized by cyanin, the exposure required may vary from ten minutes to an hour. To get the best results, the hinder surface of A should be blackened, and the front surface of B should be thrown out of action by the superposition of a wedge-shaped plate of glass, the intervening space being filled with oil of turpentine or other fluid having nearly the same refraction as glass. Moreover, the light should be purified from blue rays by a trough containing solution of bichromate of potash. With these precautions the dark parts of the bands are very black, and the exposure may be prolonged much beyond what would otherwise be admissible.

By this method it is easy to compare one flat with another, and thus, if the first be known to be free from error, to determine the errors of the second. But how are we to obtain and verify a standard? The plan usually followed is to bring three surfaces into comparison. The fact that two surfaces can be made to fit another in all azimuths proves that they are spherical and of equal curvatures, but one convex and the other concave, the case of perfect flatness not being excluded. If A and B fit one another, and also A and C, it follows that B and C must be similar. Hence, if B and C also fit one another, all three surfaces must be flat. By an extension of this process the errors of three surfaces which are not flat can be found from a consideration of the interference bands which they present when combined in three pairs.

The free surface of undisturbed water is almost ideally flat, and, as Lord Rayleigh (Nature, 1893, 48, 212) has shown, there is no great difficulty in using it as a standard of comparison. Following the same idea we may construct a parallel plate by superposing a layer of water upon mercury. If desired, the superior reflecting power of the mercury may be compensated by the addition of colouring matter to the water.

Haidinger’s Rings dependent on Obliquity.—It is remarkable that the well-known theoretical investigation, undertaken with the view of explaining Newton’s rings, applies more directly to a different system of rings discovered at a later date.

The results embodied in equations (1) to (8) have application in the first instance to plates whose surfaces are absolutely parallel, though doubtless they may be employed with fair accuracy when the thickness varies but slowly.

We have now to consider t constant and ′ variable in (1). If ′ be small,

and since the differences of are proportional to 2, the law of formation is the same as for Newton’s rings, where ′ is constant and t proportional to the square of the distance from the point of contact. In order to see these rings distinctly the eye must be focused, not upon the plate, but for infinitely distant objects.

The earliest observation of rings dependent upon obliquity appears to have been made by W. von Haidinger (Pogg. Ann., 1849, 77, p. 219; 1855, 96, p. 453), who employed sodium light reflected from a plate of mica (e.g. 0.2 mm. thick). The transmitted rays are the easier to see in their completeness, though they are necessarily somewhat faint. For this purpose it is sufficient to look through the mica, held close to the eye and perpendicular to the line of vision, at a sheet of white paper or card illuminated by a sodium flame. Although Haidinger omitted to consider the double refraction of the mica and gave formulae not quite correct for even singly refracting plates, he fully appreciated the distinctive character of the rings, contrasting Berührungsringe und Plattenringe. The latter may appropriately be named after him. Their tardy discovery may be attributed to the technical difficulty of obtaining sufficiently parallel plates, unless it be by the use of mica or by the device of pouring water upon mercury. Haidinger’s rings were rediscovered by O. R. Lummer (Wied. Ann., 1884, 23, p. 49), who pointed out the advantages they offer in the examination of plates intended to be parallel.

The illumination depends upon the intensity of the monochromatic source of light, and upon the reflecting power of the surfaces. If R be the intensity of the reflected light we have from (7)

$1⁄R$ = 1 + $1.6(1 − e^{2})^{2}⁄4e^{2} sin^{2}$;

from which we see that if e = 1 absolutely, 1/R = R = 1 for all values of. If e = 1 very nearly, R = 1 nearly for all values of for which sin2 is not very small. In the light reflected from an extended source, the ground will be of full brightness corresponding to the source, but it will be traversed by narrow dark lines. By transmitted light the ground, corresponding to general values of the obliquity, will be dark, but will be interrupted by narrow bright rings, whose position is determined by sin = 0. In permitting for certain directions a complete transmission in spite of a high reflecting power (e) of the surfaces, the plate acts the part of a resonator.

There is no transparent material for which, unless at high obliquity, e approaches unity. In C. Fabry and A. Pérot’s apparatus the reflections at nearly perpendicular incidence are enhanced by lightly silvering the surfaces. In this way the advantage of narrowing the bright rings is attained in great measure without too heavy a sacrifice of light. The plate in the optical sense is one of air, and is bounded by plates of glass whose inner silvered surfaces are accurately flat and parallel. The outer surfaces need only ordinary flatness, and it is best that they be not quite parallel to the inner ones. The arrangement constitutes a spectroscope, inasmuch as it allows the structure of a complex spectrum line to be directly observed. If, for example, we look at a sodium flame, we see in general two distinct systems of narrow bright circles corresponding to the two D-lines. With particular values of the thickness of the plate of air the two systems may coincide so as to be seen as a single system, but a slight alteration of thickness will cause a separation.

It will be seen that in this apparatus the optical parts are themselves of extreme simplicity; but they require accuracy of construction and adjustment, and the demand in these respects is the more severe the further the ideal is pursued of narrowing the rings by increase of reflecting power. Two forms of mounting are employed. In one instrument, called the interferometer, the distance between the surfaces—the thickness of the plate—is adjustable over a wide range. In its complete development this instrument is elaborate and costly. The actual measurements of wave-lengths by Fabry and Pérot were for the most part effected by another form of instrument called an étalon or interference-gauge. The thickness of the optical plate is here fixed; the glasses are held up to metal knobs, acting as distance-pieces, by adjustable springs, and the final adjustment to parallelism is effected by regulating the pressure exerted by these springs. The distance between the surfaces may be 5 or 10 mm.

The theory of the comparison of wave-lengths by means of this apparatus is very simple, and it may be well to give it, following closely the statement of Fabry and Pérot (Ann. chim. phys., 1902, 25, p. 110). Consider first the cadmium radiation treated as a standard. It gives a system of rings. Let P be the ordinal number of one of these rings, for example the first counting from the centre. This integer is supposed known. The order of interference at the centre will be p = P +. We have to determine this number , lying ordinarily between 0 and 1. The diameter of the ring under