due to divergence in the direction of the incident rays, and less rapid attenuation in other directions. In any plane C'P', therefore, the effect of the reflected light is negligible in the immediate neighbourhood of the point C', and would be most perceptible at points farthest removed from C'*, On the other hand, the fluctuations of intensity due to the diffraction of the direct rays are most marked in the neighbourhood of C', that is, for the smallest values of 0. We should accordingly expect to find that when d is not zero, the first few bands are practically identical in position with those due to simple diffraction, and those following are due to simple interference between the direct and the reflected rays. The formulæ given above satisfy both of these requirements. For it is obvious from the manner in which they have been deduced that they satisfy the second requirement. The first requirement is also satisfied, as, by putting small, the formulæ reduce to nλ=2d02 and x'=2d0; or, in other words, a'=2nd, for the minima of illumination, which is also the usual approximate diffraction formula. Accordingly, the complete formulæ nλ=2d02+2a03 and x'=2d0+3a02/2 would (on eliminating 0) give the posi tions of the minima over the entire field with considerable accuracy.

14. The statements made in the preceding paragraph are, however, subject to an important qualification. The validity of the formula obtained rests on the basis that, for large values of d, the positions of the minima of illumination are given by the simple relation a'=√2nd. This, however, is only an approximation, as the accurate values are to be found from Schuster's formula (see Table II., above), when the effect of the reflected light is negligible. When d is so large that the formulæ nλ=2d02+2a03 and x' = 2d0+3a02/2 give nearly the same positions for the minima as the simple relation 2ndλ, they should therefore cease to be strictly valid. The actual positions of the minima for such values of d should agree more closely with those given by Schuster's formula, and should, when d is very large, agree absolutely with the same. This qualification is, however, of importance only with reference to the first two or three bands obtained for fairly large values of d. The differences in respect of the other bands would be negligibly small.

=

15. To test the foregoing results, measurements were made of the widths of the bright bands for a series of values of d

* Debye's formula (loc. cit.) shows that the intensity of the reflected light becomes very small as approaches π.

by

up to 2 cm. Table III. shows the observed values, the values calculated from my formulæ, and the values according to Schuster's formula (which would be valid for a sharp diffracting edge in the same position). To calculate the positions of the minima given by the relations nλ=2d02+2a03, and x'=2d0+3a02/2, the first equation was solved for Horner's method, and the resulting values substituted in the second equation. The measurements of the width of the first band were rather rough on account of the indefiniteness of its outer edge. The agreement between the observed widths and the widths calculated from my formulæ is seen to be fairly satisfactory for values of d up to 3 mm. For larger values of d the observed widths agree more closely with those calculated from Schuster's formula, as explained in paragraphs 11 and 14 above.

Theory of the Fringes between the edge and the source of light.

16. As already remarked in paragraph 7, the direct and the reflected pencils tend to separate into distinct parts of the field when the focal plane of the observing microscope is Fig. 3.

put forward so as to lie between the edge of the cylinder and the source of light. Why this is so will be readily understood on a reference to fig. 3. The rays reflected

a=1.54 cm.

λ=6562 × 10-8 cm.

from the surface when produced backwards would touch the enveloping surface which lies within the cylinder. This surface, which is virtually the caustic of the reflected rays, terminates at the edge C of the cylinder, and when the focal plane of the observing microscope is moved forward from CP to a position P'C'P, in front of the edge, the boundary of the field on the right-hand side would shift into the region. of the shadow, and would, in fact, lie on the surface of the caustic at the point P1. If the focal plane P'C'P1 is considerably forward of PC, the field is seen divided into two parts. The first part P'C' consists of the direct rays alone (the reflected rays meeting P'C' being too oblique to enter into the field of the microscope), and should obviously be bounded at C' by a few diffraction-fringes of the ordinary Fresnel type. The second part of the field P,C' is due to the reflected. rays alone, and requires separate consideration.

17. In the case considered above, that is, when the focal plane is considerably in advance of the edge, the fringesystem within the shadow due to the reflected light is of the same type as that found by Airy in his well-known investigation on the intensity of light in the neighbourhood of a caustic. For the elementary pencils into which the reflected rays may be divided up diverge from points lying along the caustic, and if the point P1 at which the focal plane intersects the caustic is sufficiently removed from the edge C at which the latter terminates, Airy's investigation becomes fully applicable, but not otherwise. The rays emerging from the point P1 after passage through the objective of the microscope become a parallel pencil, while pencils emerging from points on either side of P, become convergent and divergent respectively. The reflected wave-front after passage through the objective has thus a point of inflexion on either side of which it may be taken to extend indefinitely, provided the arc CP, be long enough. Assuming the focal length to be f and the equation of the wave-front to be An3, the value of A may be readily found. The equation of the caustic is

(4x2+4y2 — a2)3 — 27a1x2=0.

From this, or directly by an approximate treatment, it may readily be shown that the radius of curvature of the caustic at the point C is the radius of the cylinder. For our present purpose, it is thus sufficient to treat the caustic as equivalent to a cylinder of radius 3a/4 touching the reflecting

where

dn2

6 dn3 3 6 dn\dn2,

is the measure of the convergence or divergence

of the normals to the wave-front in the neighbourhood of the point of inflexion. Substituting the values obtained from the formulæ of geometrical optics, it is found that

The illumination in the fringe-system alongside the caustic is then given by Airy's formula

1 being the distance of any point in the focal plane measured from the point of intersection with the caustic. The integral gives a series of maxima of which the first is the largest, and the rest gradually converge to zero. The minima of illumination are zeroes*. As the focal plane is moved further and further towards the source of light, the fringe-system moves inwards along the caustic, but remains otherwise unaltered.

18. The foregoing treatment of the reflected fringe-system in terms of Airy's theory ceases to be valid when the focal plane is not sufficiently in advance of the edge, and the arc CP of the caustic is therefore not large enough. For the reflected wave-front on one side of the point of inflexion then becomes limited in extent, and its equation cannot with sufficient accuracy be assumed to be of the simple form

An3, extending to infinity in either direction. In fact, when the focal plane is at the edge of the cylinder and CP1 is zero, the point of inflexion coincides with the extreme edge of the reflected wave-front. At this stage, of course, the

Graphs of Airy's integral and references to the literature will be found in an interesting paper by Aichi and Tanakadate (Journal of the College of Science, Tokyo, vol. xxi. Art. 3).