may be regarded as approximately parallel, and the preceding investigation will apply.

Let R be the radius of the lens, O the point of contact, and let OM = p. Then PM D, and

=

(2R-D) D = p2,

whence neglecting D2, we have D= p2/2R, accordingly

8 = 2πp2/Rx.cos r..

and the reflected light vanishes when

8 = 2nπ or p2 = nRλ sec r.........

At 0, p=0 and therefore 8=0; whence the central spot is black. If homogeneous light is employed, the central spot will be surrounded by a series of dark rings, whose diameters are proportional to the square roots of the natural numbers.

The intensity will be a maximum, when

S=(2n+1)π or p2 = (n + 1) Rλ secr..

..(10);

accordingly there will be a series of bright rings whose diameters are proportional to √ √ √ etc.

29

Since the diameters of the rings are also proportional to (sec r), it follows that the rings increase as the angle of incidence increases.

For light of different colours, the diameters of the rings vary as λ; consequently when sunlight is employed, a number of coloured rings are observed.

The inner edge of the first ring is dark blue, and its outer edge red; and the order of succession of the colours of the first seven rings was found by Newton to be as follows:-(1) black, blue, white, yellow, red; (2) violet, blue, green, yellow, red; (3) purple, blue, green, yellow, red; (4) green, red; (5) greenish-blue, red; (6) greenish-blue, pale red; (7) greenish-blue, reddish-white. This

COLOURS OF THICK PLATES.

33

list is usually known as Newton's scale of colours; and the expression "red or blue of the third order," refers to the colour of that name seen in the third ring.

In the preceding discussion of Newton's rings, we have supposed that a thin stratum of air constitutes the thin plate; consequently it is possible to increase the angle of incidence until it exceeds the critical angle. Under these circumstances, it will be found that there is still a system of coloured rings, and that the central spot is black; but the consideration of this question must be deferred, until we have discussed the dynamical theory of reflection and refraction.

The system of transmitted rings is complementary to the reflected system, but is less distinct.

Colours of Thick Plates.

29. The phenomenon known as the colours of thick plates was first observed by Newton', who allowed sunlight, proceeding into a darkened room through a hole in the window-shutter, to fall perpendicularly upon a concave mirror formed of glass quicksilvered at the back. A white opaque card pierced with a small hole was placed at the centre of curvature of the mirror, so that the regularly reflected light returned through the small hole, and a set of coloured rings was observed on the card surrounding the hole. The Duc de Chaulnes on repeating this experiment, observed that the brilliancy of the rings was much increased by spreading over the surface of the mirror a mixture of milk and water, which was allowed to dry, and thus produced a permanent tarnish. The colours of thick plates were first explained on the undulatory theory by Young, who attributed them to the interference of two streams of light, one of which is scattered on entering the glass, and then regularly reflected and refracted, whilst the other is regularly reflected and refracted, and then scattered on emerging from the first surface; but the complete explanation is due to Stokes, which we shall now consider.

1 Optics, Book II. part 4.

2 Mém. de l'Académie, 1755, p. 136.

On the Colours of Thick Plates," Trans. Camb. Phil. Soc. vol. ix. p. 147.

B. O.

3

30. Stokes' investigation is based on the following hypothesis:

In order that two streams of scattered light may be capable of interfering, it is necessary that they should be scattered, in passing and repassing, by the same set of particles. Two streams, which are scattered by different sets of particles, although they may have come originally from the same source, behave with respect to each other like two streams coming from different sources.

It will hereafter be proved, that if this law were not true, it would follow, that if a luminous point were viewed through a plate of glass, both of whose surfaces were tarnished with milk and water, coloured rings would be seen; but on performing the experiment no rings were observed. Moreover Stokes calculated the retardation of the stream scattered on emergence relatively to that scattered at entrance, and found that the dimensions of the rings were such that they could not possibly have escaped notice had they been formed. This experiment is decisive, but the truth of the law is also apparent from theoretical considerations; for the dimensions of particles of dust, although small compared with the standards of ordinary measurement, are not small in comparison with the wave-length of light, so that the light scattered at entrance taken as a whole is most irregular; and the only reason why regular interference is possible at all is, that each particle acts twice in a similar manner, once when the wave enters and again when it emerges.

31. We shall now work out the problem when the mirror is plane.

M.

T

Let L be the luminous point, E the eye of the observer; let Lo, E, be the feet of the perpendiculars let fall from Land E on to the dimmed face of the mirror.

Let LSTPE be the course of the ray, which is regularly refracted and reflected at S and T, and scattered on emergence at P; and let LPVQE be the course of the ray, which is scattered entering the glass and is then regularly reflected and refracted.

=

Let LPs, EP-u, LL = c, EE, h; also let t be the thickness of the plate, μ its index of refraction, i, r the angles of incidence and refraction at S; R1, R2 the retardations of the rays LSTPE and LPVQE. Then

and

R1 = LS +2μST + PE

=c sec i + 2μt secr+ (h2 + u2)1......(11),

c tan i + 2t tan r=s, sini =μ sin r....

..(12).

Now experiment shows, that in order to see the rings distinctly the angle of incidence must be small, whence i, r, s and u are small quantities. We may therefore, as a sufficient approximation, neglect powers of small quantities above the second. Expanding in powers of i, r and u, we obtain from (11)

R1 = c + 2μt +h+ } (ci3 + 2μtr2 + u2/h)............ (13).

and

Again, if ', ' be the angles of incidence and refraction at Q,

The intensity of the light entering the eye is therefore pro

portional to cos2 πR/λ.

Let E, be the origin, and let EE be the axis of 2, and let the plane az pass through L. Let x, y be the coordinates of P, and let E.L. = a.

Also let the thickness of the plate be supposed to be so small, that its square may be neglected. Then substituting in (16), we obtain

For a given fringe R is constant; hence the fringes form a system of concentric circles, whose common centre lies on the axis Hence if a be the abscissa of the centre

of x.

Now ah (h+c) and ah/(h — c) are the abscissæ of the points, in which the plane of the mirror is cut by two lines drawn from the eye to the luminous point and its image respectively. We thus obtain the following construction for finding the centre of the system:―Join the eye with the luminous point and its image, and produce the former line to meet the mirror; then the middle point of the line joining the two points, in which the mirror is cut by the two lines joining the eye, will be the centre of the system.

Hence if the luminous point be placed to the right of the perpendicular let fall from the eye on to the plane of the mirror, and between the mirror and the eye, the concavity of the fringes will be turned to the right. If the luminous point, still lying on the right, be now moved backwards, so as to come beside the eye and ultimately fall behind it, the curvature will decrease until the fringes become straight, after which it will increase in the