This will be more easily understood by the inspection of a figure, which represents a section of the two mirrors and of the reflected undulations, formed by a plane drawn from the luminous point perpendicularly to the mirrors represented by DE and DF. The luminous point is supposed to be S, and A and B are the geometrical positions of its two images, which are determined by the perpendiculars SA and SB falling from S on the mirrors, taking in them PA = SP [p132] and QB = SQ. The points A and B, thus found, are the centres of divergence of the rays reflected from the respective mirrors, according to the well known law of reflection. Thus, in order to have the direction of the ray reflected at any point G of the mirror DF, for example, it is sufficient to draw a right line through B and G, which will be the direction of the reflected ray. Now it must be remarked, that, according to the construction by which the position of B is found, the distances BG and SG will be equal, and thus the whole route of the ray coming from S and arriving at b, is the same as if it had come from B. This geometrical truth being equally applicable to all the rays reflected by the same mirror, it is obvious that they will arrive at the same instant at all the points of the circumference n′bm, described on the point B as a centre, with a radius equal to Bb; consequently this surface will represent the surface of the reflected undulation when it arrives at b, or, more correctly speaking, its intersection with the plane of the figure: the surface of the undulation being understood as relating to the points which are similarly agitated at the same instant: the points being all, at the commencement of the whole oscillation, for example, or at the middle or the end, completely at rest; and in the middle of each semioscillation, possessed of the maximum of velocity.
In order to represent the two systems of reflected undulations, there are drawn, with the points A and B for their centres, two different series of equidistant arcs, separated from each other by an interval which is supposed equal to the length of a semiundulation. In order to distinguish the motions in opposite directions, the arcs on which the motions of the ethereal particles are supposed to be direct, are represented by full lines, and the maximum of the retrograde motions are indicated by dotted lines. It follows that the intersections of the dotted lines with the full lines are points of complete discordance, and of course show the middle of the dark stripes; and, on the contrary, the intersections of similar arcs show the points of perfect agreement, or the middle of the bright stripes. The intersections of the arcs of the same kind are joined by the dotted lines b′p′, br, b′p′, and those of arcs of [p133] different kinds by the full lines n′o′, no, no, n′o′: these latter representing the successive positions or the trajectories of the middle points of the dark stripes, and the former the trajectories of the bright bands.
It has been necessary to magnify very greatly in this figure the real length of the luminous undulations, and to exaggerate the mutual inclination of the two mirrors, so that we must not expect an exact representation of the phenomenon, but merely a mode of illustrating the distribution of the interferences, in undulations which cross each other with a slight inclination.
It is easy to deduce from geometrical considerations, that the length of these fringes is in the inverse ratio of the magnitude of the angle made by the two pencils which interfere, and that the interval, comprehended between the middle points of two consecutive dark or bright bands, is as much greater than the length of the undulation, as the radius is greater than the sine of the angle of intersection.
In fact the triangle bni, formed by the right line bi, and the two circular arcs ni and nb, may be considered as rectilinear and isosceles, on account of the smallness of the arcs; and the sine of the angle bni, considered as very small, may be called ibbn: so that bn being the radius, ib will represent the sine of the angle bni, which has its legs perpendicular to those of the angle AbB: consequently, these angles being equal, one of them may be substituted for the other; and representing by i the angle AbB, formed by the reflected rays, we have bn = ibsin i; consequently nn, which is twice bn, will be equal to 2ibsin i. But nn is the distance between the middle points of two consecutive dark stripes, and is the distance which has been called the breadth of a fringe; and ib being the breadth of a semiundulation, according to the construction of the figure, 2ib will be that of a whole undulation; consequently the breadth of a fringe may be said to be equal to the length of an undulation divided by the [numerical] sine of the angle made by the reflected rays [p134] with each other, which is also the angle under which the interval AB would appear to an eye placed at b. We find another equivalent formula, by remarking that the two triangles, bni and AbB, are similar, whence we have the proportion bn: bi = Ab : AB, and bn = bi × AbAB, or 2bn = 2bi × AbAB: which implies that we may find the numerical breadth of a fringe by multiplying the length of an undulation by the distance of the images A and B from the plane on which the fringes are measured, and dividing the product by the distance of the two images.
It is sufficient to inspect the figure, in order to be convinced of the necessity of having the two mirrors nearly in the same plane, if we wish to obtain fringes of tolerably large dimensions; for in the little triangle bni, the side bi, which represents the length of a semiundulation, being little more than the hundred thousandth of an inch for the yellow rays, for example, the side bn, which measures the half breadth of a fringe, can only become sensible when bn is very little inclined to in, so that their intersection may be remote from ib; and the inclination of bn to in depends on the distance AB, which is the measure of the inclination of the mirrors.
If A and B, instead of being the images of the luminous point, were the projections of two very fine slits cut in a screen RN, through which the rays of light were admitted from a luminous point placed behind the screen in the continuation of the line bDC, the two paths described between the point and the slits A and B being equal, it would be sufficient to compute the paths described by the rays, beginning from A and B, in order to have the differences of their lengths; and it is obvious in this case, that the calculations which we have been making of the breadth of the fringes, produced by the two mirrors, would remain equally applicable, at least as long as each slit remained narrow enough to be considered as a single centre of undulation, relatively to the inflected rays which it transmits. It may therefore be said that the breadth of the fringes, produced by two very fine slits, is equal to the length of an undulation supposed [p135] to be multiplied by the interval between the two slits, and divided by the distance of the screen from the wires of the micrometer employed for measuring the fringes.
This formula is also applicable to the dark and bright stripes which are observed in the shadow of a narrow substance, substituting the breadth of this substance for the interval which separates the two slits, as long as the stripes are far enough from the edges of the shadow: for when they approach very near to the edges, it is shown, both by theory and by experiment, that this calculation does not represent the facts with sufficient accuracy; and it is not perfectly correct in all cases, either for the fringes within the shadow, or for those of the two slits, but only for the fringes produced by the mirrors, which exhibit the simplest case of the interference of rays slightly inclined to each other. In order to obtain from the theory, a rigorous determination of the situation of the dark and light stripes in the two former cases, it is not sufficient to calculate the effect of two systems of undulations, but those of an infinite number of similar groups must be combined, according to a principle which will shortly be explained, in treating of the general theory of diffraction.
ii. Rule for the Correction of a LUNAR OBSERVATION. By Mr. WILLIAM WISEMAN, of Hull. [◊]
RULE.