If $ be the breadth of a fringe, which may be measured by a micro meter, /3= A f A= d The distances a, 6, to the former of which a' may be deemed equal, are not small, and can be measured without difficulty. The distance d was measured by Freenel by placing a screen with a small round hole at a known distance from the mirrors, so that a slender beam of light from each of the images 1, e passed through the hole, and measuring by a micrometer the distance between the centres of the beams at a known distance on the other side of the hole whence d is obtained from the measured distance by similar triangles. This is one method of measuring the length of a wave of light.
The excessive smallness of A, indicated by this or any similar pheno menon of interference, leads to a complete explanation of one of the oldest difficulties belonging to the undulatory theory, the existence of rays and shadows. Conceive a broad beam of light to fall perpen dicularly on a screen containing a moderately small aperture, and let ua examine the disturbance produced at a point m, situated at a con siderable distance on the other side of the screen. For greater simplicity we shall suppose the incident beam to come from a very distant point, so that the incident waves may be regarded as plane. By Huygens's principle each element of the front of a wave, as it arrives at the piano of the aperture, may be considered as the centre of an elementary disturbance which diverges into tho space behind the screen, and in duo time reaches Y. But the disturbance at M will be by no means proportional to the size of the aperture, since the various secondary waves which at a given instant reach m, and which arise from incident waves which reach in succession the plane of the aperture, are in a condition to interfere. First, suppose to situated at some distance outside the geometrical projection of the aperture. Make m the centre of a number of spheres with radii increasing by t A, and of which as many are drawn as cut the aperture. These spheres will cut the aperture into numerous narrow slips, of the form of portions of circular annuli, having for their common centre the projection N of the point am on the plane of the aperture. It will be readily seen that the aggregate effect of the secondary waves starting from the various elements of one slip will be as nearly as possible neutralised by that of the waves coining from the next slip. For the squares of the radii of the annuli Increase in arithmetical progression, and therefore the areas of consecutive slips are equal, except as to the trifling difference due to the change in the angle subtended at N. Neglecting for the moment
this small change, we readily ace that to each element of the first slip corresponds an equal clement of the second, at a distance from m different from that of the former element by tA, and therefore the secondary waves belonging to these two elements will, as nearly as pomible, neutralise each other's effect. hence the joint effect of two consecutive slips as compared with that of either of them is a small quantity of the order A, that is the ratio of A to the other quantities involved, such as the difference of distance from at of opposite sides of the aperture. But as the number of slips is a large quantity of the order it might be supposed that the total effect was comparable with that of one slip. This however ie not the case. For the effect of any slip taken along with half the effects of the two adjacent slips is a small quantity of time order A=, and the sum of all such, when we group the slips so as to take every alternate slip along with half the effect of its two neighbours, is only a small quantity of the order A, unless it be in consequence of the want of compensation at the beginning and end of the series. But at the two ends, that is at the parts of the aperture nearest to and farthest from the point N, the of the slip. de indica down to zero. The peculiar case in which a part of the boundary of the aperture is exactly circular, having its centre in N, which attaches itself to the theory of the bright point in the centre of the shadow of a circular disc, is supposed to be excluded from con sideration. We infer therefore that the disturbance at a point RI, situated as above described, is insensible.
Neat suppose m to lie at sonic distance inside the geometrical pro jection of the boundary of the aperture. Imagine a series of spheres drawn as before around m, beginning with that which touches the plane of the aperture iu the point N. The aperture will now be cut spas before, only that now for some dietaries, round N the annuli will be complete, after which they will become incomplete, and will finally dwindle away to nothing. The neutralisation will in this case take place just as before, except an regards half the effect of the tint annulus, or central circle of the syatein, and the disturbance will therefore be sensibly the same as if the screen wore removed.