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 from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflection. It must, however, be borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflection is more oblique.

The formula expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory. Let A0B0 be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface A0B0 to A or B is determined by the condition that the optical distance, ∫ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus

We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface A0B0 to the point A is changed; but in virtue of the minimum property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A0B0 to A is represented by ∫( + )ds, the integration being along the original path A0 A; and similarly the optical distance between A0B0 and B is represented by ∫( + )ds, the integration being along B0  B. In virtue of (4) the difference of the optical distances to A and B is

The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispersing substance, ∫ ds＝.s, where s is simply the thickness traversed by the ray. If t2 and t1 be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion is given by ＝(t2 − t1)/a, or, if t1 be negligible,

The condition of resolution of a double line whose components subtend an angle is that  must exceed /a. Hence, in order that a double line may be resolved whose components have indices and  +, it is necessary that t should exceed the value given by the following equation:—

 8. Diffraction Gratings.—Under the heading “Colours of Striated Surfaces,” Thomas Young (Phil. Trans., 1802) in his usual summary fashion gave a general explanation of these colours, including the law of sines, the striations being supposed to be straight, parallel and equidistant. Later, in his article “Chromatics” in the supplement to the 5th edition of this encyclopaedia, he shows that the colours “lose the mixed character of periodical colours, and resemble much more the ordinary prismatic spectrum, with intervals completely dark interposed,” and explains it by the consideration that any phase-difference which may arise at neighbouring striae is multiplied in proportion to the total number of striae.

The theory was further developed by A. J. Fresnel (1815), who gave a formula equivalent to (5) below. But it is to J. von Fraunhofer that we owe most of our knowledge upon this subject. His recent discovery of the “fixed lines” allowed a precision of observation previously impossible. He constructed gratings up to 340 periods to the inch by straining fine wire over screws. Subsequently he ruled gratings on a layer of gold-leaf attached to glass, or on a layer of grease similarly supported, and again by attacking the glass itself with a diamond point. The best gratings were obtained by the last method, but a suitable diamond point was hard to find, and to preserve. Observing through a telescope with light perpendicularly incident, he showed that the position of any ray was dependent only upon the grating interval, viz. the distance from the centre of one wire or line to the centre of the next, and not otherwise upon the thickness of the wire and the magnitude of the interspace. In different gratings the lengths of the spectra and their distances from the axis were inversely proportional to the grating interval, while with a given grating the distances of the various spectra from the axis were as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of wave-lengths, and the method of separating the overlapping spectra by a prism dispersing in the perpendicular direction. He described also the complicated patterns seen when a point of light is viewed through two superposed gratings, whose lines cross one another perpendicularly or obliquely. The above observations relate to transmitted light, but Fraunhofer extended his inquiry to the light reflected. To eliminate the light returned from the hinder surface of an engraved grating, he covered it with a black varnish. It then appeared that under certain angles of incidence parts of the resulting spectra were completely polarized. These remarkable researches of Fraunhofer, carried out in the years 1817–1823, are republished in his Collected Writings (Munich, 1888).

The principle underlying the action of gratings is identical with that discussed in § 2, and exemplified in J. L. Soret’s “zone plates.” The alternate Fresnel’s zones are blocked out or otherwise modified; in this way the original compensation is upset and a revival of light occurs in unusual directions. If the source be a point or a line, and a collimating lens be used, the incident waves may be regarded as plane. If, further, on leaving the grating the light be received by a focusing lens, e.g. the object-glass of a telescope, the Fresnel’s zones are reduced to parallel and equidistant straight strips, which at certain angles coincide with the ruling. The directions of the lateral spectra are such that the passage from one element of the grating to the corresponding point of the next implies a retardation of an integral number of wave-lengths. If the grating be composed of alternate transparent and opaque parts, the question may be treated by means of the general integrals (§ 3) by merely limiting the integration to the transparent parts of the aperture. For an investigation upon these lines the reader is referred to Airy’s Tracts, to Verdet’s Leçons, or to R. W. Wood’s Physical Optics. If, however, we assume the theory of a simple rectangular aperture (§ 3); the results of the ruling can be inferred by elementary methods, which are perhaps more instructive.

Apart from the ruling, we know that the image of a mathematical line will be a series of narrow bands, of which the central one is by far the brightest. At the middle of this band there is complete agreement of phase among the secondary waves. The dark lines which separate the bands are the places at which the phases of the secondary wave range over an integral number of periods. If now we suppose the aperture AB to be covered by a great number of opaque strips or bars of width d, separated by transparent intervals of width a, the condition of things in the directions just spoken of is not materially changed. At the central point there is still complete agreement of phase; but the amplitude is diminished in the ratio of a : a + d. In another direction, making a small angle with the last, such that the projection of AB upon it amounts to a few wave-lengths, it is easy to see that the mode of interference is the same as if there were no ruling. For example, when the direction is such that the projection of AB upon it amounts to one wave-length, the elementary components neutralize one another, because their phases are distributed symmetrically, though discontinuously, round the entire period. The only effect of the ruling is to diminish the amplitude in the ratio a : a + d; and, except for the difference in illumination, the appearance of a line of light is the same as if the aperture were perfectly free.

The lateral (spectral) images occur in such directions that the projection of the element (a + d) of the grating upon them is an exact multiple of. The effect of each of the n elements of the grating is then the same; and, unless this vanishes on account of a particular adjustment of the ratio a : d, the resultant amplitude becomes comparatively very great. These directions, in which the retardation between A and B is exactly mn, may be called the principal directions. On either side of any one of them the illumination is distributed according to the same law as for the central image (m＝0), vanishing, for example, when the retardation amounts to (mn ± 1). In considering the relative brightnesses of the different spectra, it is therefore sufficient to attend merely to the principal directions, provided that the whole deviation be not so great that its cosine differs considerably from unity.

We have now to consider the amplitude due to a single element, which we may conveniently regard as composed of a transparent part a bounded by two opaque parts of width d. The phase of the resultant effect is by symmetry that of the component which comes from the middle of a. The fact that the other components have phases differing from this by amounts ranging between ± am/(a + d) causes the resultant amplitude to be less than for the central image (where there is complete phase agreement).