WAVE THE O II Y But, if ^ = ixat and -cos))xdxdt/ . The occurrence of the anomalous factor in (6) is thus explained. It must be admitted that the present process of investigation is rather artificial ; and the cause lies in the attempt to dispense with the differential equation satisfied by i^, viz., on which in the case of sound the whole theory is based. It is in fact easy to verify that any value of ^ included under (8), where P 9 -($-aO s +(n-y) s + 2 i satisfies the equation "When there is no question of resolution by Huygens s principle, the distinction between |, 75 and x, y may be dropped. Starting from the differential equation, we may recover previous results very simply. If if be proportional to cospx cosqy, we have f + ( 2 -F ! -<f) / = ..... (11). If K--2 --q- = n~, M being real, the solution of (11) is where A and B are independent of volving t, x, y, we may write Restoring the factors in of which the first term may be dropped when we contemplate waves travelling in the positive direction only. The corresponding realized solution is of the type $ = cos 2)x cosqy cos {icat- (/c 2 -^ 2 - q-).z] . . (13). When K" > (p- + q 2 ), the wave travels without change of type and with velocity r _ K(t // ! _ _ />2 (**) We have now to consider what occurs when K-<(p" write /c 2 -2^ 2 -2 2 = -M 2 > we have in place of (12) and for the realized solution corresponding to (13) l/ = cos px cos qyc~ tiz cos If we (15); (16). We conclude that under these circumstances the motion rapidly diminishes as z increases, and that no wave in the usual sense can be propagated at all. >orruga- It follows that corrugations of a reflecting surface (no matter how .ions of deep) will not disturb the regularity of a perpendicularly reflected period wave, provided the wave-length of the corrugation do not exceed ess that of the vibration. And, whatever the former wave-length may han A be in relation to the latter, regular reflexion will occur when the lie out. incidence is sufficiently oblique. The first form of solution may be applied to give an explanation shadow of the appearances observed when a plane wave traverses a parallel >f a coarse grating and then impinges upon a screen held at varying Crating, distances behind. 1 As the general expression of the wave periodic with respect to x in distance a we may take sin(i<at -/*i where 2> = 27r/a- , K = 27T/A , and ^ = K? - p~ , fj..f = /c 2 - -ip 2 , . . . , the series being continued as long as /j. is real. We shall here, however, limit ourselves to the first three terms, and in them sup pose A! and B x to be small relatively to A . The intensity may then be represented by AO^ + 2 A Aj cos (px +/) cos (K~ - /uj-) + 2A B 1 coa(px+g)ain( K z-n 1 z) .... (17). The stripes thrown upon the screen in various positions are thus periodic functions of s, and the period is if X be supposed small in comparison with <r. It may bo noticed that, if the position of the screen be altered by the half of this amount, the effect is equivalent to a shifting parallel to x through the distance ^<r. Hence, if the grating consists of alternate trans parent and opaque parts of width 4<r, the stripes seen upon the screen are reversed when the latter is drawn back through the dis tance O--/A. In this case we may suppose B x to vanish, and (17) 1 Phil. Mag., March 1881, "On Copying Diffraction Gratings and on Some Phenomena Connected Therewith." then shows that the field is uniform when the screen occupies positions midway between those which give the most distinct patterns. These results are of interest in connexion with the photographic reproduction of gratings. 16. Talbot s Bands. These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, half the aperture being covered by a thin plate, e.g., of glass or mica. The view of the matter taken by the dis coverer 2 was that any ray which suffered in traversing the plate a retardation of an odd number of half wave-lengths would be extinguished, and that thus the spectrum would be seen inter rupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago s theory of stellar scintillation. 3 It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a half-wave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of the bands comes about in a very curious way, as is shown by a circumstance first observed by Brewster. When the retarding plate is held on Brew- the side towards the red of the spectrum, the bands are not seen, ster s ob- Even in the contrary case, the thickness of the plate must not serva- exceed a certain limit, however pure the spectrum may be. A tions. satisfactory explanation of these bands was first given by Airy, 4 but we shall here follow the investigation of Stokes, 5 limiting ourselves, however, to the case where the retarded and unretarded beams are contiguous and of equal width. The aperture of the unretarded beam may thus be taken to be limited by x= -h, x = Q, y = -I, y= +1 , and that of the beam retarded by R to be given by x = 0, x = h, y = - I, y +1. For the former (1) 11 gives (1), "j ~~ -j j on integration and reduction. For the retarded stream the only difference is that we must sub tract R from at, and that the limits of x are and + &. We thus get for the disturbance at |, rj due to this stream / . Krf If . K& . f at _ f _ E+ ^ (2) _ If we put for shortness T for the quantity under the last circular function in (1), the expressions (1), (2) may be put under the forms ?(siiiT, ^sin(r a) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin T and cos T in the expression u sin T + v sin (T - a) , so that I = M 2 + v 2 + 2uv cos a , which becomes on putting for u, v, and o their values, and putting Q. (3), . . (4). If the subject of examination be a luminous line parallel to ?j, we shall obtain what we require by integrating (4) with respect to ?j from oo to+ oo . The constant multiplier is of no especial interest, so that we may take as applicable to the image of a line If R= JA, I vanishes at | = ; but the whole illumination, repre- / + oo sented by/ I d , is independent of the value of R. If R = 0, in agreement with 11, where a has the meanin 1 ~ 2ir|/4 --J5S1U- A/ ~, here attached to 1h. The expression (5) gives the illumination at | due to that part of the complete image whose geometrical focus is at = 0, the retardation for this component being R. Since we have now to integrate for the whole illumination at a particular point due to all the components which have their foci in its neighbourhood, we
- Phil. Mag., x. p. 3C4, 1837.
3 On account of inequalities in the atmosphere giving a variable refraction, the light from a star would be irregularly distributed over a screen. The experiment is easily made on a laboratory scale, with a small source of light, the rays from which, in their course towards a rather distant screen, are disturbed by the. neighbourhood of a heated body. At a moment when the eye, or object-glass of a telescope, occupies a dark position, the star vanishes. A fraction of a second later the aperture occupies a bright place, and the star reappears. According to this view the chromatic effects depend entirely upon atmospheric dispersion. Phil. Trans., 1840, p. 225; 1841, p. 1. * Phil. Trans., 1848, p. 227.
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