431 posed to be infinitely small, and two radiant points, however near together, form separated images. This is tantamount to an assumption that A is infinitely small. The actual finiteness of A imposes a limit upon the separating or resolving power of an optical instrument. This indefiniteness of images is sometimes said to be due to diffraction by the edge of the aperture, and proposals have even been made i or curing it by causing the transition between the interruptfd and transmitted parts of the primary wave to be less abrupt. Such a view of the matter is altogether misleading. AVhat requires explanation is not the imperfection of actual images so much as the possibility of their being as good as we find them. At the focal point (| = 0,ij = 0) all the secondary waves agree in phase, and the intensity is easily expressed, whatever be the form of the aperture. From the general formula (2), if A be the area of aperture, To 2 ~ A 2 /A;/" 3 (7). The formation of a sharp image of the radiant point requires that the illumination become insignificant when |, rj attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture. So long as there is no sensible discrepancy of phase there can be no sensible diminution of brightness as compared with that to be found at the focal point itself. We may go further, and lay it down that there can be no considerable loss of brightness until the difference of phase of the waves pro ceeding from the nearest and furthest parts of the aperture amounts to ^A. When the difference of phase amounts to A, we may expect the resultant illumination to be very much reduced. In the par ticular case of a rectangular aperture the course of things can be readily followed, especially if we conceive / to be infinite. In the direction (suppose horizontal) for which rj = 0, |//=sin0, the phases of the secondary waves range over a complete period when sin0 = A/, and, since all parts of the horizontal aperture are equally effective, there is in this direction a complete compensation and consequent absence of illumination. When sin0 = |A/, the phases range one and a half periods, and there is revival of illu mination. W r e may compare the brightness with that in the direc tion = 0. The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion + -- / _ cos<t>d<f> : 1, or - : 1 ; so that the brightness in this direction is pr^r of the oir 9ir- maximum at 0=-0. In like manner we may find the illumination iu any other direction, and it is obvious that it vanishes when sin & is any multiple of A/a. The reason of the augmentation of resolving power with aper ture will now be evident. The larger the aperture the smaller are the angles through which it is necessary to deviate from the principal direction in order to bring in specified discrepancies of phase the more concentrated is the image. L,umin- In many cases the subject of examination is a luminous line of ms line, uniform intensity, the various points of which are to be treated as independent sources of light. If the image of the line be = 0, the intensity at any point {, 17 of the diffraction pattern may be represented by tf the same law as obtains for a luminous point when horizontal directions are alone considered. The definition of a fine vertical line, and consequently the resolving power for contiguous vertical lines, is thus independent of the vertical aperture of the instrument, a law of great importance in the theory of the spectroscope. The distribution of illumination in the image of a luminous line is shown by the curve ABC (fig. 5), representing the value of the function siu-u/u~ from u = Q to u = 2-n: The part corresponding to negative values of u is similar, OA being a line of symmetry. Let us now consider the distribution of brightness iii the image of a double line whose components are of equal strength, and at such an angular interval that the central line in the image of one coincides with the first zero of brightness in the image of the other. In fig. 5 the curve of brightness for one component is ABC, and for the other OA C ; and the curve representing half the combined brightnesses is E BE. The brightness (corresponding to B) mid way between the two central points AA is 8106 of the brightness at the central points themselves. We may consider this to be about the limit of closeness at which there could be any decided appearance of resolution, though doubtless an observer accustomed to his instrument would recognize the duplicity with certainty. The obliquity, corresponding to U = TT, is such that the phases of the secondary waves range over a complete period, i.e., such that the projection of the horizontal aperture upon this direction is one wave-length. We conclude that a double line cannot be fairly resolved unless its components sub tend an angle exceeding that sub tended by the ivavc-lcngth of light at a distance equal to the horizontal aperture. This rule is convenient on account of its simplicity ; and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolu tion. On the expe- limental confir- 71 "f mation of the theory of the resolving power of rectangular apertures, see OPTICS, vol. xvii. p. 807. If the angular interval between the components of a double line be half as great again as that supposed in the figure, the brightness midway between is 1802 as against 1 0450 at the central lines of each image. Such a falling off in the middle must be more than sufficient for resolution. If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance equal to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images. Since the limitation of the width of the central band in the image of a luminous line depends upon discrepancies of phase among the secondary waves, and since the discrepancy is greatest for the waves which come from the edges of the aperture, the ques tion arises how far the operation of the central parts of the aperture is advantageous. If we imagine the aperture reduced to two equal narrow slits bordering its edges, compensation will evidently be complete when the projection on an oblique direction is equal to vj-A, instead of A as for the complete aperture. By this procedure the width of the central band in the diffraction pattern is halved, and so far an advantage is attained. But, as will be evident, the bright bands bordering the central band are now not inferior to it in brightness ; in fact, a band similar to the central baud is repro duced an indefinite number of times, so long as there is no sensible discrepancy of phase in the secondary waves proceeding from the various parts of the same slit. Under these circumstances the narrowing of the band is paid for at a ruinous price, and the arrangement must be condemned altogether. A more moderate suppression of the central parts is, however, sometimes advantageous. Theory and experiment alike prove that a double line, of which the components are equally strong, is better resolved when, for example, one-sixth of the horizontal aperture is blocked off by a central screen; or the rays quite at the centre may be allowed to pass, while others a little further removed are blocked off. Stops, each occupying one-eighth of the width, and with centres situated at the points of trisectiou, answer well the required purpose. It has already been suggested that the principle of energy requires that the general expression for P in (2) when integrated over the whole of the plane {, TJ should be equal to A, where A is the area of the aperture. A general analytical verification has been given by Stokes. 1 The expression for P may be written in the form Central stops. Total intensity propor tional to aperture. the integrations with respect to x , y 1 as well as those with respect to x, y being over the area of the aperture ; and for the present purpose this is to be integrated again with respect to |, rj over the whole of the focal plane. In changing the order of integration so as to take first that with respect to , rj, it is proper, in order to avoid ambiguity, to intro duce under the integral sign the factor e^^P* 1 , the + or - being chosen so as to make the elements of the integral vanish at infinity. After the operations have been performed, a and & are to be supposed to vanish. Thus_// r I L W|t^ = Limit of 7 - y) }
1 Ed. Trans., xx. p. l>17,