present paper. With a sharp edge, the fringes of the Fresnel type disappear when the focal plane coincides with it, and reappear without alteration of type when the focal plane is between the edge and the source of light. As mentioned above and shown in figs. VII. and VIII. of the Plate, fringes of this type may also be observed with the cylinder when the focal plane is in this position, and in addition we have, inside the shadow, an entirely separate system of fringes characterized by perfectly black minima and a series of maxima with intensities converging to zero. This latter system has nothing in common with the diffraction phenomena of the Fresnel class, and has obviously an entirely different origin. That it is formed exclusively by the light reflected from the surface of the cylinder is proved by the fact that it may be cut off without affecting the rest of the field by screening the surface. It is accordingly clear that the light reflected from the surface of the cylinder plays a most important part in the explanation of the phenomena, and that the edge of the cylinder grazed by the incident rays alone acts as a diffracting edge in the usual way, and not all the elements of the surface as supposed by Brush. We shall accordingly proceed on this basis to consider the theory of the fringes observed in various positions of the focal plane of the objective.

Theory of the Fringes at the edge of the cylinder.

9. When the focal plane coincides with the edge at which the incident light grazes the cylinder, it is permissible to regard the fringes seen as formed by simple interference between the light that passes the cylinder unobstructed and the light that suffers reflexion at the surface of the cylinder at various incidences; for, if a sharp diffracting edge be put in the focal plane in the same position, no diffraction-fringes would be visible. The positions of the minima of illumination in the field may be readily calculated.

In fig. 2, let O be the centre of the cross-section of the cylinder in the plane of incidence, and let C be the point at which the light grazes the cylinder. It is sufficient for practical purposes to consider the incident beam as a parallel pencil of rays. The ray meeting the cylinder at the point Q is reflected in the direction QP. Let QOA=0, so that OQP=+0, and <OPQ=—- - 20. Let a be the radius of the cylinder and CP=x. The difference of path, 8, between the direct ray and the reflected ray reaching the

2

point P is evidently equal to QP-RP, which can be easily shown to be given by

Therefore, neglecting 4th and higher powers of 0, we have

so that

8=2a03, and x=3a02/2,

8=2a).

3a

Since by reflexion the rays suffer a phase change of half a wave-length, the edge (will form the centre of a dark band, and the successive minima are therefore given by

where n=1, 2, 3, &c. The results calculated according to the above theory and those found in experiment are given in Table I.

error.

The discrepancies are within the limits of experimental When making these measurements, the focal plane was, in the first instance, set in approximate coincidence with the edge of the cylinder by noting the stage at which a further movement of the focal plane towards the light results in a movement of the fringes into the region of the shadow. There was, however, a slight uncertainty in regard to this adjustment, and the best position of the focal plane was finally ascertained by actual trial.

10. The ratio between the maxima and minima of illumination in the fringes at the edge may readily be calculated. Dividing up the pencil of rays incident on the cylinder into elements of width a sin edo or aede approximately, the width of the corresponding elements of the reflected pencil in the plane of the edge is da, that is, 3a0 de. The amplitude of the disturbance at any point in this plane due to the reflected rays is thus only 1/3 of that due to the direct rays, multiplied by the reflecting power of the surface. If the reflecting power be unity (as is practically the case at such oblique incidences), the ratio of the intensities of the maxima and minima is (1 + 1/3)/(1-1/v3)2, that is approximately 14: 1. The dark bands are thus nearly, but not quite, perfectly black.

Theory of the Fringes at the edge of the shadow.

11. If the fringes be observed in a plane (such as C'P' in fig. 2) which is farther from the source of light than the edge of the cylinder, the diffraction and mutual interference of the direct and the reflected rays have both to be taken into account. Since the reflected rays form a divergent

pencil while the incident rays are parallel, the effect of the former at any point sufficiently removed from the cylinder would be negligible in comparison with the effect of the latter. If d, the distance of the plane of observation from the edge of the cylinder, be sufficiently large, the problem thus practically reduces to one of simple diffraction of the incident waves by the straight edge C. The positions of the minima of illumination with reference to the geometrical edge of the shadow would then be given approximately by the simple formula

x'=√/2ndλ=√/4n√/dx/2,

where 'C'P' and d=CC',

or with great accuracy by Schuster's formula,
x' = √ (8n−1)dx/4= √(8n−1)/2√/dx/2.

The two formulæ give results which do not differ materially except in regard to the first two or three bands, as can be seen from Table II.

12. If d be not large, the intensity of the reflected rays is not negligible. The following considerations enable us to find a simple formula for the positions of the minima of illumination which takes both diffraction and interference into account. We may, to begin with, find the positions of the minima assuming the case to be one of simple interference between the direct and the reflected rays. The expression for the path difference, d', of the rays arriving at the point P' is readily seen from fig. 2 to be given by the formula

d' = (d+a sin )(sec 20—1).

Also,

Ꮎ

x'd tan 20+ a(cos 0 sec 20−1).

=

These two relations may, to a close approximation, be written in the form

Putting d=0, we get the formula already deduced (see paragraph 9 above) for the fringes at the edge of the cylinder. On the other hand, if d be greater than a, we may, to a sufficient approximation, write

and the positions of the points at which the direct and the reflected rays are in opposite phases are given by the formula

x' =√2ndλ.

13. But, as remarked above, the simple formula x' = √2ndλ also gives the approximate positions of the minima in the diffraction-fringes at a considerable distance from the cylinder, where the effect of the reflected rays is negligible. It is thus seen that the formulæ

and

nλ=2d02+2a03,

x'. =2d0+3a02/2,

}

(A)

suffice to give the approximate positions of the minima of illumination at the edge of the cylinder (at which point the fringes are due to simple interference of the direct and the reflected rays) and also at a considerable distance from it (in which case they are due only to the diffraction of the incident light). A priori, therefore, it would seem probable that the formulæ would hold good also at intermediate points, that is for all values of d. That this is the result actually to be expected may be shown by considering the effect due to the reflected rays at various points in the plane of observation. The reflected wave-front is the involute of the virtual caustic (see fig. 3 below). At the edge C, the radius of curvature of the wave-front is zero, and increases rapidly as we move outwards from the edge of the cylinder. The reflected rays accordingly suffer the most rapid attenuation