When two plane mirrors are placed with their reflecting surfaces towards each other, and parallel, they form the experiment called the endless gallery. Let (in fig. 4) the arrow, Q, be placed vertically between the parallel mirrors, CD, BA, with their sil vered faces turned to one another, Q will produce in the mirror CD the image q'. This image will act as a new object to produce with the mirror BA. the image q', which, again, will produce with the mirror CD another image, and so on. Another series of images, such as q', q", etc., will similarly be produced at the same time, the first of the series being q', the image of Q in the mirror BA. By an eye placed between the mirrors, the succession of images will be seen as described; and if the mirrors were perfectly plane and parallel, and reflected all the light incident on them, the num ber of the images of both series would be infinite. If, instead of being parallel, the mirrors are inclined at an angle, the form and position of the image of an object may be found in precisely the same way as in the former case, the image formed with the first mirror being regarded as a new (virtual) object, whose image, with regard to the second, has to be determined. For a curious application of two plane mirrors meeting and inclined at an angle an aliquot part of 180°, see art. KALEIDOSCOPE.-3. The two propositions already established are of extensive application, as has partly been shown. They include the explanation of all phenomena of light related to plane mirrors. The thir1 proposition is one also of considerable utility, though not fundamental. It is: When a ray of light has been reflected at each of two mirrors inclined at a given angle to each other, in a plane perpendicular to their intersection, the reflected ray will deviate from its original course by an angle double the angle of inclination of the mirrors. Let A and 13 (fig. 5) be sections of the mirrors in a plane perpendicular to their intersection, and let their directions be produced till they meet in C. Let SA, in the plane of A and B, be the ray incident on the first mirror at A, and let AB be the line in which it is thence reflected to B. After reflection at B, it will pass in the line BD, meeting SA, its original path, produced in D. The angle AD13 evidently measures its deviation from itK original course, and this angle is readily shown to be double of the angle at C, which is that of the inclination of the mirrors. It is on this proposition that the, important. mathematical instruments called the quadrant and sex tant (q.v.) depend Carted Srnface8.—As when a pencil of light is reflected by a curved mirror, each ray follows the ordinary law of reflection, in every case in which we can draw the normals for the different points of the surface, we can determine the direction in which the various rays of the pencil are reflected, as in the case of plane mirrors. It so happens that normals can be easily brawn only in Alm case of the sphere, and of a few "surfaces of revolution," as they are called. These are the paraboloid, the ellipsoid, and the hyperboloid of revolution. The paraboloid of revolution is of importance in optics, as it is used in some specula for telescopes. See arts. SPECULUM and TELE SCOPE. The three surfaces last named are, however, all of them interesting, as being for pencils of light incident in certain ways what are called surfaces of accurate reflec tion—i.e., they reflect all the rays of the incident pencil to a single point or focus. We shall explain to what this property is owing in the case of the parabolic reflector, and state generally the facts regarding the other two.
1. The concave parabolic reflector is a surface of accurate reflection for pencils of rays parallel to the axis or central line of figure of the paraboloid. This results from the property of the surface, that the normal at any point of it passes through the axis, and bisects the angle between a line through that point, parallel to the axis, and a line joining the point to the focus of the generating parabola. Referring to fig. 6, suppose a ray incident on the surface at P, in the line SP, parallel to the axis AFG. Then if F be the focus of the generating parabola, join PF. PF is the direction of the reflected ray. For PG, the normal at P, by the property of the surface, bisects the angle FPS, and therefore Z (angle) FPG= LGPS. But SPG is the angle of incidence, and SP, PG, and FP are in one plane, and, therefore, by the laws of reflection, FP is the reflected ray. In the same way, all rays whatever, parallel to the axis, must pass through F after reflection. If F were a luminous point, the rays from it, after reflec tion on the mirror, would all proceed in a cylindrical pencil parallel to the axis. This reflector, with a bright light in its focus, is accordingly of common use in light houses.
2. In the concave ellipsoid mirror there are two points—viz., the foci of the gener ating ellipse, such that rays diverging from either will be accurately reflected to the other. This results from the property of the figure, that the normal at any point bisects the angle included between lines drawn to that point from the foci.
3. Owing to a property of the surface similar to that of the ellipsoid, a pencil of rays converging to the exterior focus of a hyperbolic reflector, will be accurately reflected to the focus of the generating hyperbola.
The converse of the above three propositions holds in the case of the mirrors being convex.
Though the sphere is not a surface of accurate reflection, except for rays diverging from the center, and which on reflection are returned thereto, the spherical reflector is of great practical importance, because it can be made with greater facility and at less expense than the parabolic reflector. See art. TELESCOPE. It is necessary, then, to investigate the phenomena of light reflected from it.
4. is usual to treat of two cases, the one the more frequent in practice, the other the more general and comprehensive in theory. First, then, to find the focus of reflected rays when a small pencil of parallel rays is incident directly on a concave spherical mirror. Let BAB' (fig. 7) be a section of the mirror, 0 its center of curvature, and A the center of its aperture. AO is the axis of the mirror, and there fore of the incident pencil, because it is incident directly on the mirror; a pencil being called oblique when its axis is at an angle to the axis of the mirror. As the ray inci dent in the line OA will be reflected back in the same line—OA being the normal at A— the focus of reflected rays must be in OA. Let SP be one of the rays; it will be reflected so that L qPO= L SPO. But L POq= L OPS by parallel lines, Therefore, Z qPO= L and Pq and Oq are equal. If, now, the incident pencil be very small—i.e., if P be very near A—then the line Pq will very nearly coincide with the line OA. and Pq and Oq will each of them become very nearly the half of OA. Let F be the middle point of ON—the point, namely, to which q tends as the pencil diminishes. The F is called the principal focus of the mirror, and AF the principal focal length, which is thus = radius of the mirror. It will be observed that when AP is not small, q lies between A and F. Fq is called the aberration of the ray. When AP is large, the reflected rays will continually intersect, and form a luminous curve with a cusp at F. This curve is called the caustic We shall now proceed to the more general case of a small pen cil of diverging rays, incident directly on a concave spherical mirror. Let PAP' (fig. 8) be a section of the mirror, A the center of its aperture, 0 of its curvature, and let F be its principal focus. Then, if Q be the focus of incident rays (as if _proceeding from • a candle there situated), q, the focus of the reflected rays, lies on QOA, since the pencil is incident directly, and the ray QOA, being incident in the line of the normal OA, is reflected back in the same line. Let PQ be any other ray of the pencil. It will be reflected in Pg, so that Z gP0= L OPQ; and on the supposition that PA is very small, so that QP becomes nearly equal to QA, and qP to gA, it can be shown, by Euclid, vi. 3, that very nearly. From this equation is deduced the formula gA. AF which enables us to find qA, when QA and AF are known. Thus, let the radius of vature be 12 in., and the distance of the source of the rays, or QA, 30 in., the focal length gA = 30X 6 = 7 inches. If the rays bad diverged upon g, it is clear they — would have been reflected to Q. The points Q and g, accord ingly, arc called conjugate foci.
If the mirror be convex, as in fi;. 9 instead of concave, and a pencil of diverging rays be incident directly on it from 4' Q, we should find, proceeding in exactly the same way as in Q A the former case, the equation Aq = x ,' or taking the O el A Q 30 X 6 same numbers as before; qA = 30 + 6 = 5 inches.
For information regarding the formation of images by Fig. 9. spherical mirrors, the reader may consult Potter's Elenients of Optics. See also the arts. Minnons and IMAGES.
By considering fig. 8, it is easy to see how the relative positions of the two con jugate foci, as they arc called, Q and q, vary as the distance, AQ, of the origin of the rays is changed. As Q is advanced towards 0, g also approaches 0, since the angles QPO and qP0 always remain equal; and when the source of the light is in the center, 0, of the sphere, the reflected rays are all returned upon the source. As Q, again, recedes from 0, g moves towards F, which it does not quite reach until the distance of Q is infinite, so that the incident rays may be considered as parallel, as in fig. 7. If Q is placed between 0 and F, then q will be to the right of 0; and when Q coincides with F, the reflected rays will have no focus, but will be parallel. If Q is between F and A, the reflected rays will diverge, and will have their virtual focus to the left of A. The correctness of these deductions may easily be verified. The positions of the conjugates are traced in precisely the same way for the convex mirror, and the reader who is inter ested will find no difficulty in tracing them for himself.