termined by the difference of their distances from that point, or (which comes to the same thing) by the difference of the lengths of the lines connecting it with the edges of the obstacle. The phenomena of interference are therefore the same as in the case of light emanating from two near origins, already considered; and we may transfer to the present case the conclusions arrived at in (81). Accordingly, if e denote the breadth of the obstacle, and 6 its distance from the screen, the distance, x, of any band from the centre of the shadow is

(103) The positions of the fringes formed by a narrow rectangular aperture are determined by a similar formula.

Let PP' be the section of the aperture, PAP' the portion of the wave which has just reached it, diverging from the luminous origin at O; and let QQ' be the projection of the aperture on the screen. Then, if we take the point R on this screen in such a manner, that the difference of its distances from the edges of the aperture, RP RP, shall

be equal to a whole

number of semi-undula

A

R

છુ

B

P

tions, that point will be the centre of a dark or bright band, according as the assumed number is even or odd. For, in the former case, the wave PAP' may be divided into an even number of parts, such that the distances of every two consecutive points of division from the point R differ by half an undulation; the waves sent by every two consecutive portions to the point R will therefore be in complete discordance, and the total effect at that point will be null. On the other hand, when the difference RP' - RP is equal to an odd number of semi-undulations, the number of opposing portions of the wave will be odd, and as the alternate portions compensate

each other's effects at the point R, there will remain one portion producing there its full effect.

The successive bands being formed at the points for which RP' - RP = nλ, it is obvious that their distances, RB, from the centre of the projection of the aperture, will be given by the same formula as in the case last considered, c being now the breadth of the aperture,-with this difference, however, that the dark bands correspond to the even values of n, and the bright bands to the odd values, which is the reverse of what takes place in the bands formed within the shadow of an opaque obstacle. We learn then, 1st, that the distances of the successive fringes of any colour form an arithmetical progression whose common difference is equal to its first term; 2ndly, that they vary directly as the distance of the screen, and inversely as the breadth of the aperture; and 3rdly, that they are proportional to the length of the wave; and therefore greatest for the extreme red rays, least for the extreme violet, and of intermediate magnitude for the rays of intermediate refrangibility.

We have supposed the screen to be so remote that the bands are entirely without the projection of the aperture. This will obviously be the case when QP - QP is less than half a wave. When the distance of the screen is so small that QP-QP exceeds this limit, fringes will be visible also within the projection of the aperture. In this case the portions into which the wave is divided are sensibly different in magnitude, as well as obliquity. The reasoning above employed is therefore no longer applicable; and the points of maximum and minimum brightness can only be obtained by a complete calculation of the intensity of the light.

(104) The phenomena of diffraction hitherto considered are of the simplest class but as such phenomena arise in every instance in which light is in part intercepted, it is obvious that they admit of endless modifications, varying with

the form of the interposed body. Some of these are too remarkable to pass unnoticed.

Among the most striking of these effects are those produced by light diverging from a luminous origin, and transmitted through a small circular aperture ;-as, for example, that formed by a pin in a sheet of lead. When the transmitted light is viewed through a lens, the image of the aperture appears as a brilliant spot, surrounded by coloured rings of great vividness; and these vary in the most beautiful manner, as the distance of the aperture from the luminous origin, or from the eye, is altered. When the latter distance is considerable, the central spot is white, and the coloured rings follow the order observed in thin plates. As the eye approaches the aperture, the central white spot contracts to a point, and then vanishes. The rings then close in on it in order; and the centre assumes in succession the most vivid and beautiful hues, altogether similar to those of the reflected rings of thin plates.

This remarkable coincidence has been shown to be an exact result of theory. It has been demonstrated that the intensity of the light of any simple colour, at the central spot, -and the compound tint in the case of white light,—will be the same as that reflected from a plate of air, whose thickness bears a certain simple relation to the radius of the aperture, and its distances from the luminous origin and from the eye.

The points of maximum and minimum intensity are easily determined.

Let O be the luminous point, and OAB the line drawn from it through the centre of the aperture PP'; then the interval of retardation, d, of

the ray which reaches any point B on this line, coming from the edge of the aperture, is OP+ PB-OB. Let OA= a,

AB=b, and AP = r; then, since r is very small in comparison with a and b, it is easy to see that

Now when this interval is equal to a whole number, n, of semi-undulations, the aperture may be divided by concentric circles, such that the rays which reach the point B, coming from any two successive circumferences, shall differ by the interval of half a wave. It follows from the preceding formula that the squares of the radii, and therefore the superficies of the successive circles thus formed, are as the numbers of the natural series; so that the annuli comprised between every two succeeding circumferences are equal. But the elementary waves proceeding from each annulus are in complete discordance with those from the two adjacent. The successive annuli will therefore destroy one another's effects, and the total intensity of the light at the point B will be null, or equal to that of the last, according as the number of annuli (the central circle included) is even or odd. Hence, for a given aperture, there will be a succession of points on the axis, at which the intensity of the light is alternately nothing and a maximum ; and it is obvious from the preceding that the distances of these points will be the values of b given by the formula

in which the points of complete darkness correspond to the even values of n, and those of maximum brightness to the odd values.

Such is the case with homogeneous light. As the points of maximum and minimum intensity are different for the rays of different colours, there will be no point of complete darkness in compound light, but a succession of points, at which the centre of the aperture is richly coloured.

(105) The theory of Fresnel is not only in exact accordance with facts already known: it has also led to many new and unexpected conclusions, and predicted consequences which have been afterwards verified on trial. One of the most remarkable of these is the phenomenon of diffraction by an opaque circular disc. Poisson applied Fresnel's integrals to this case; and he was led to the startling result, that the illumination of the centre of the shadow was precisely the same as if the disc had been altogether removed. The principles already laid down will enable the reader to satisfy himself of the theoretical truth of this conclusion. Arago was the first to show that it was in accordance with fact, and his experiment may be repeated without much difficulty.

(106) We have seen that when light diverging from a luminous point passes by the edges of a fine hair or wire, a succession of coloured bands will be formed parallel to the edge of the shadow; and the distances of these bands from the shadow, and from one another, will be greater, the less the diameter of the wire. If many such wires be exposed to the diverging beam, and if, instead of being parallel, they are crossed and interlaced in every possible direction, it is easy to conceive that the coloured bands will be disposed in concentric circles, whose centre is the luminous point. These circles resemble the halos visible round the Sun and Moon in hazy weather. Their diameters vary in the inverse ratio of the thickness of the wires or fibres.

This law was applied by Young, in a very ingenious manner, to the comparison of the diameters of fibres, or small particles of any kind.

A plate of metal is perforated with a small round hole, about theth of an inch in diameter, around which, at the distance of about or of an inch, is a circle of smaller holes. The flame of a lamp is then placed immediately behind the aperture, and the luminous point viewed through the substance to be