442 W A V E T H E O 11 Y may conveniently regard as origin. | is then the coordinate relatively to of any focal point for which the retardation is R ; and the required result is obtained by simply integrating (5) with respect to from oo to + oo . To each value of | corresponds a different value of A, and (in consequence of the dispersing power of the plate) of R. The variation of A may, however, be neglected in the integration, except in 2?rR/A, where a small variation of A entails a comparatively large alteration of phase. If we write p = 27rR/A ....... (6), we must regard p as a function of |, and we may take with sufficient approximation under any ordinary circumstances p = p -r*r ........ (7), where p denotes the value of p at 0, and -sris a. constant, which is positive when the retarding plate is held at the side on which the blue of the spectrum is seen. The possibility of dark bands depends upon TZ being positive. Only in this case can cos{ p + (TV - 27r/f/A/)| } retain the constant value - 1 throughout the integration, and then only when
- r = 27rA/A/ ....... (8),
and cosp = -1 ........ (9). The first of these equations is the condition for the formation of dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory. The integration can be effected without much difficulty. For the first term in (5) the evaluation is effected at once by a known formula. In the second term if we observe that = cos p cos g^ H- sin p sin g^ , we see that the second part vanishes when integrated, and that the remaining integral is of the form - (10). where By differentiation with respect to g l it may be proved that w = from g 1 = - =o to g 1 = - 2h : , w = ^ir(2h 1 + g 1 ) from g 1 =-2h 1 to ^ = 0, w = ^Tr(2h l -g 1 ) from ^=0 to a 1 = 2h 1 , i# = from fj^ 2/ij to r/, = oo . The integrated intensity, 1 , or 2irh 1 + 2 cos p w , is thus when g 1 numerically exceeds 2/tjj and, when g 1 lies between 2h v I = 7r{2/i 1 + (2/t 1 -Vt/i 2 )cosp } .... (12). Best It appears therefore that there are no bunds at all unless & lies thick- between and + 4A 1; and that within these limits the best bands are ness. formed at the middle of the range when w=2/; 1 . The formation of bands thus requires that the retarding plate bo held upon the side already specified, so that -a be positive ; and that the thickness of the plate (to which & is proportional) do not exceed a certain limit, which we may call 2T . At the best thickness T the bauds are black, and not otherwise. The linear width of the band (c) is the increment of | which alters p by 2ir, so that c = 2ir/vr ........ (13). With the best thickness W=27T//A/ ....... (14), so that in this case e = f/h ........ (15). Width of The bands are thus of the same width as those due to two infinitely bauds, narrow apertures coincident with the central lines of the retarded and uuretarded streams, the subject of examination being itself a fine luminous line. If it be desired to see a given number of bands in the whole or in any part of the spectrum, the thickness of the retarding plate is thereby determined, independently of all other considerations. Experi- But in order that the bands may be really visible, and still more mental in order that they may be black, another condition must be satisfied. condi- It is necessary that the aperture of the pupil be accommodated tions. to the angular extent of the spectrum, or reciprocally. Black bands will be too line to be well seen unless the aperture (2/0 of the pupil be somewhat contracted. One-twentieth to one-fiftieth of an inch is suitable. The aperture and the number of bands being both fixed, the condition of blackness determines the angular magni tude of a band and of the spectrum. The use of a grating is very convenient, for not only are there several spectra in view at the same time, but the dispersion can be varied continuously by sloping the grating. The slits may be cut out of tin-plate, and half covered by mica or " microscopic glass," held in position by a little cement. If a telescope be employed there is a distinction to be observed, Use of according as the half-covered aperture is between the eye and telescope. the ocular, or in front of the object-glass. In the former case the function of the telescope is simply to increase the dispersion, and the formation of the bands is of course independent of the par ticular manner in which the dispersion arises. If, however, the half-covered aperture be in front of the object-glass, the pheno menon is magnified as a whole, and the desirable relation between the (unmagniiied) dispersion and the aperture is the same as with out the telescope. There appears to be no further advantage in the use of a telescope than the increased facility of accommodation, and for this of course a very low power suffices. The original investigation of Stokes, here briefly sketched, ex- More tends also to the case where the streams are of unequal width general h, k, and are separated by an interval 2y. In the case of unequal investiga- width the bands cannot be black ; but if h = k, the iiniteness of tion of 2y does not preclude the formation of black bands. Stokes. The theory of Talbot s bands with a half-covered circular aperture has been treated by H. Struve. 1 17. Diffraction when the Source of Light is not Seen in Focus. The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail ; but, in view of their historical interest and of the case with which many of the experiments may be tried, some account of their theory could not be excluded from such a work as the present. One or two examples have already attracted o in attention when considering Huygens s zones, viz., the shadow of a circular disk, and of a screen circularly perforated ; but the most famous problem of this class first solved by Fresnel relates to the shadow of a screen bounded by a straight edge. In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the plane passing through the luminous point and perpendicular to the dif fracting edges, supposed to be straight and parallel. In strictness this idea is appropriate only when the source is a luminous line, emitting cylindrical waves, such as might be obtained from u luminous point with the aid of a cylindrical lens. When, in order to apply Huygens s principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner ( 15) to that of the element which lies in the plane of reference, and may be considered to be repre sented thereby. The same method of representation is applicable to spherical waves, issuing from a point, if the radius of curvature be large ; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practi cally upon that of the central parts where the phase is sensibly constant. 2 In fig. 21 APQ is the arc of the circle representa tive of the wave-front of resolution, the centre being at 0, and the radius OA being equal to a. B is the point at which the effect is required, distant a + b from 0, so that AB = &; Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by cos 2? where S = BP-AP is the retardation relatively to that from A. Now t $ }.ds . r J at B of the wave from so that, if we write the effect at B is $ ab n 2jr5 A" ir(a ab (1), t -2irt r 2irt /" ) < cos / cosirv.dv + su / sin^nv-.dv i (3) ( T J T J ) the limits of integration depending upon the disposition of the diffracting edges. When a, b, A are regarded as constant, the first factor may be omitted, as indeed should be done for consistency s sake, inasmuch as other factors of the same nature have been omitted already. The intensity I 2 , the quantity with which we are principally concerned, may thus be expressed 1 St Petersburg Trans., xxxi., No. 1, 1SS3. 2 In experiment a line of li^ht is sometimes substituted for a point in order to increase the illumination. The various parts of the line are here independent sources, and should be treated accordingly. To assume a cylindrical form of primary wave would be justifiable only when there is synchronism among the
secondary waves issuing from the various centres.