438 spending to m = Q not being counted as a spectrum), if the grating interval cr or (a + d) is less than A. Under these circumstances, it the material of the grating be completely transparent, the whole of the light must appear in the direct image, and the ruling is not perceptible. From the absence of spectra Frauuhofer argued that there must be a microscopic limit represented by A ; and the infer ence is plausible, to say the least. 1 Fraunhofer should, however, have fixed the microscopic limit at , as appears from (5), when we suppose 6 = ^ir, <j> ^ir. Resolving We will now consider the important subject of the resolving V?wer. power of gratings, as dependent upon the number of lines (?i) and the order of the spectrum observed (m). Let BP (fig. 11) be the direction of the principal maximum (middle of central band) for the wave-length A in the ??i th spectrum. Then the relative retardation of the extreme rays (corre sponding to the edges A, B of the grating) is mn. If BQ be the direction for the first minimum (the darkness between the central and first lateral band), the rela tive retardation of the extreme rays is (mn + l). Suppose now that A + 8A is the wave-length for which BQ gives the principal maximum, then whence SA/A = l/mn ....... (6). According to our former standard, this gives the smallest difference of wave-lengths in a double line which can be just resolved ; and we conclude that the resolving power of a grating depends only upon the total number of lines, and upon the order of the spectrum, without regard to any other considerations. It is here of course assumed that the n lines are really utilized. In the case of the D-lines the value of SA/A is about 1/1000 ; so that to resolve this double line in the first spectrum requires 1000 lines, in the second spectrum 500, and so on. It is especially to be noticed that the resolving power does not depend directly upon the closeness of the ruling. Let us take the case of a grating 1 inch broad, and containing 1000 lines, and consider the effect of interpolating an additional 1000 lines, so as to bisect the former intervals. There will be destruction by inter ference of the first, third, and odd spectra generally ; while the advantage gained in the spectra of even order is not in dispersion, nor in resolving power, but simply in brilliancy, which is increased four times. If we now suppose half the grating cut away, so as to leave 1000 lines in half an inch, the dispersion will not be altered, while the brightness and resolving power are halved. There is clearly no theoretical limit to the resolving power of gratings, even in spectra of given order. But it is possible that, as suggested by Rowland, 2 the structure of natural spectra may be too coarse to give opportunity for resolving powers much higher than those now in use. However this may be, it would, always be possible, with the aid of a grating of given resolving power, to construct artificially from white light mixtures of slightly different wave-length whose resolution or otherwise would discriminate between powers inferior and superior to the given one. 3 If we define as the "dispersion" in a particular part of the spectrum the ratio of the angular interval dO to the corresponding increment of wave-length dA, we may express it by a very simple formula. For the alteration of wave-length entails, at the two limits of a diffracted wave-front, a relative retardation equal to mnd. Hence, if a be the width of the diffracted beam, and d6 the angle through which the wave-front is turned, or dispersion = mn/a ....... (7). The resolving power and the width of the emergent beam fix the optical character of the instrument. The latter element must eventually be decreased until less than the diameter of the pupil of the eye. Hence a wide beam demands treatment with further apparatus (usually a telescope) of high magnifying power. In the above discussion it has been supposed that the ruling is accurate, and we have seen that by increase of m a high resolving power is attainable with a moderate number of lines. But this procedure (apart from the question of illumination) is open to the objection that it makes excessive demands upon accuracy. Accord ing to the principle already laid down, it can make but little dif ference in the principal direction corresponding to the first spectrum, provided each line lie within a quarter of an interval (a + d) from its theoretical position. But, to obtain an equally good result in the m th spectrum, the error must be less than I/TO of the above amount. 4 1 "Notes on Some Fundamental Propositions in Optics," Phil. Mag., June 18o6. 2 Compare also Lippich, Pogg. Ann., cxxxix. p. 4G5, 1870; Rayleigh, Nature, Oct. 2, 1 873. 3 The power of a grating to construct light of nearly definite wave-length is well illustrated by Young s comparison with the production of a musical note by reflexion of a sudden sound from a row of palings. The objection raised by Herschel (Light, 703) to this comparison depends on a misconception. It must not be supposed that errors of this order of magnitude are unob jectionable in all cases. The position of the middle of the bright band represen- There are certain errors of a systematic character which demand special consideration. The spacing is usually effected by means of a screw, to each revolution of which corresponds a large number (e.g., one hundred) of lines. In this way it may happen that, Approxi- although there is almost perfect periodicity with each revolution mate of the screw after (say) 100 lines, yet the 100 lines themselves are period- not equally spaced. The "ghosts" thus arising were first de- ieity. scribed by Quiucke, 5 and have been elaborately investigated by Peirce, 6 both theoretically and experimentally. The general nature of the effects to be expected in such a case may be made clear by means of an illustration already employed for another purpose. Suppose two similar and accurately ruled transparent gratings to be superposed in such a manner that the lines are parallel. If the one set of lines exactly bisect the intervals between the others, the grating interval is practically halved, and the previously exist ing spectra of odd order vanish. But a very slight relative dis placement will cause the apparition of the odd spectra. In this case there is approximate periodicity in the half interval, but com plete periodicity only after the whole interval. The advantage of approximate bisection lies in the superior brilliancy of the surviving spectra ; but in any case the compound grating may be considered to be perfect in the longer interval, and the definition is as good as if the bisection were accurate. The effect of a gradual increase in the interval (fig. 12) as we Gradually pass across the grating has been investigated by Cornu, 7 who thus increasin{ explains an anomaly observed by Mascart. The latter found that interval, certain gratings exercised a converging power upon the spectra formed upon one side, and a corresponding diverging power upon the spectra on the other side. Let us suppose that the light is incident perpendicularly, and that the grating interval increases from the centre towards that edge which lies nearest to the spec trum under observation, and decreases towards the hinder edge. FIG. 12. FIG. FIG. 15. It is evident that the waves from both halves of the grating are accelerated in an increasing degree, as we pass from the centre outwards, as compared with the phase they would possess were the central value of the grating interval maintained throughout. The irregularity of spacing has thus the effect of a convex lens, which accelerates the mar- ginal relatively to the central rays. On the other side the effect FlG - 16 - ry - FlG - ".-*2y. FIG. : is reversed. This kind of irregularity may clearly be present in a degree surpassing the usual limits, without loss of definition, when the telescope is focused so as to secure the best effect. It may be worth while to examine further the other variations from correct ruling which correspond to the various terms express ing the deviation of the wave-surface from a perfect plane. If x and y be coordinates in the plane of the wave-surface, the axis of y being parallel to the lines of the grating, and the origin corre sponding to the centre of the beam, we have as an approximate equation to the wave-surface ( 6) z = - (- fixy + j~, + ax 3 + $y?y + yxy* + Sy 3 + . . (8) ; ^p Zp and, as we have just seen, the term in ce 2 corresponds to a linear error in the spacing. In like manner, the term in y 2 corresponds to a Curva- general curvature of the lines (fig. 13), and does not influence the ture. definition at the (primary) focus, although it may introduce astigmatism. 8 If we suppose that everything is symmetrical on the_two sides of the primary plane y = Q, the coefficients B, 0, 8 vanish. In spite of any inequality between p and p , the definition will be good to this order of approximation, provided a and 7 vanish. The former measures the thickness of the primary focal line, and the latter measures its curvature. The error of ruling other giving rise to a is one in which the intervals increase or decrease in errors. both directions from the centre outwards (fig. 14), and it may often be compensated by a slight rotation in azimuth of the object-glass of the observing telescope. The term in 7 corresponds to a varia tion of curvature in crossing the grating (fig. 15). _ When the plane zx is not a plane of symmetry, we have to con sider the terms in xy, x-y, and y s . The first of these corresponds tative of a mathematical line can be fixed with a spider-line micrometer within a small fraction of the width of the band, just as the accuracy of astronomical observations far transcends the separating power of the instrument. 5 Pogg. Ann., cxlvi. p. 1, 1872. c Ann. Jour. Math., vol. ii. p. 330, 1879. 7 C. R., Ixxx. p. 645, 1875. " In the same way we may conclude that in flat gratings any departure from a straight lino has the effect of causing the dust in the slit and the spec trum to have different foci a fact sometimes observed" (Rowland, " On Concave
Gratings for Optical Purposes/ Phil. Mag., September 1883).