58 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 



stopped. Now the whole of the aperture AB is not effective in producing illumination in the 

 direction RS. For let C be the centre of AB, and through C draw a plane perpendicular to 

 RS, and then draw a pair of parallel planes each at a distance ^\ from the former plane, 

 cutting AB in M lt N u another pair at a distance X, and cutting AB in M. 2 , N 2 , and so on 

 as long as the points of section fall between A and B. Let M, N be the last points of section. 

 Then the vibrations proceeding from MN in the direction RS neutralize each other by inter- 

 ference, so that the effective portions of the aperture are reduced to AM, NB. Now the 

 distance between the feet of the perpendiculars let fall from A, M on RS may have any value 

 from to -i\, and for the angle of diffraction actually employed AM was equal to about twice 

 that distance on the average, or rather less. Hence AM may be regarded as ranging from 

 to X; and since for the brightest part of a band forming that portion of a spectrum of the 

 first class which belongs to light of given refrangibility AM has just half its greatest value, we 

 may suppose AM = ^X. But if the distance between the planes eh, ab be a small multiple 

 of X, and y be small, ef will be small compared with X, and therefore compared with AM. 

 Hence the breadth of the portions of the plane eh, such as ef, for which we are not at liberty to 

 regard the light as first diffracted and then regularly refracted, is small compared with the 

 breadth of the portions of the aperture, such as AM, which are really effective ; and therefore, so 

 far as regards the main part of the illumination, we are at liberty to make the supposition just 

 mentioned. But we must not suppose the wave to be first regularly refracted and then 

 diffracted, because the regular refraction presupposes the continuity of the wave. 



The above reasoning is not given as perfectly satisfactory, nor could we on the strength 

 of it venture to predict with confidence the result ; but the result having been obtained 

 experimentally, the explanation which has just been given seems a plausible way of accounting 

 for it. According to this view of the subject, the result is probably not strictly exact, but 

 only a very near approximation to the fact. For, if we suppose the distance between the 

 planes eh, ab to be only a small multiple of X, we cannot apply the regular law of refraction, 

 except as a near approximation. Moreover, the dynamical theory of diffraction points to the 

 existence of terms which, though small, would not be wholly insensible at the distance of the 

 plane ab. Lastly, when the radius of a secondary wave which passes the edge A or B is only 

 a small multiple of X, we cannot regard y as exceedingly small. 



Let us consider now the results of experiments Nos. 11 and 12. In diffraction at refraction, 

 the amount of crowding with respect to which the theory leaves us in doubt vanishes along 

 with fx — 1 ; and although this amount is far from insensible in the actual experiments, it is 

 still not sufficiently large to prevent the results from being decisive in favour of one of the two 

 hypotheses respecting the direction of vibration. Thus the curves marked "A" in the first 

 figure are well separated from those marked " E n , and if m were to approach indefinitely to 1, 

 the curves I. A and II. A would approach indefinitely to III. A, and I.E, and II. E to III. E. 

 In diffraction at reflection, however, the case is quite different, and in the absence of a precise 

 theory little can be made of the experiments, except that they tend to confirm the law ex- 

 pressed by the equation (49). In the case of the first and second images the diffraction 

 accompanied refraction, and so far the experiments were of the same nature as those which have 

 been just discussed, but the angle of incidence was not equal to zero, and in that respect they differ. 



