Page:The American Cyclopædia (1879) Volume XII.djvu/669

OPTICS 655 and size of images resolves itself into investi- gating the images of a series of points. And first, the case of a single point. A, fig. 2, placed before a plane mirror, M N, will be considered. Any ray, A B, incident from this point on the mirror, is reflected in the direction B O, ma- king the angle of reflection D B O equal to the angle of incidence D B A. If now a per- pendicular, A N, be let fall from the point A on the mirror, and if the ray O B be pro- longed below the mir- ror until it meets this perpendicular in the point a, two triangles are formed, A B N FIG. 2. and N B a, which are equal, for they have the side B N common to both, and the angles A N B, A B 1ST, equal to the angles a N B, a B N ; for the angles A N B and a N B are right angles, and the angles A B N" and a B 1ST are equal to the angle O B M. From the equality of these triangles, it follows that a N" is equal to A N ; that is, that any ray, A B, takes such a direction after being reflect- ed, that its prolongation below the mirror cuts the perpendicular A a in the point a, which is at the same distance from the mirror as the point A. This applies also to the case of any other ray from the point A, A for example. From this the important consequence follows, that all rays from the point A, reflected from the mirror, follow after reflection the same di- rection as if they had proceeded from the point a. The eye is deceived, and sees the point A at a, as if it were really situate at a. Hence in plane mirrors the image of any point is formed behind the mirror at a distance equal to that of the given point from its front surface, and on the perpendicular let fall from this point on the mirror. It is manifest that the image of any object will be obtained by constructing accord- ing to this rule the image of each of its points, . or at least of those which are sufficient to de- termine its form. Fig. 3 shows how the image FIG. 3. a 5 of any object, A B, is formed. It follows from this construction that in plane mirrors the image is of the same size as the object ; for if the trapezium A B C D be applied to the tra- pezium D a 5, they are seen to coincide, and 619 VOL. xii. 42 the object A B agrees with its image. A fur- ther consequence of the above construction is, that in plane mirrors the image is symmet- rical in reference to the object, and not invert- ed. When an object is between two plane mirrors nearly parallel, the primary images seen in each of these are reflected as if at a greater distance in the other, and so on, form- ing in each mirror a long succession of images, growing more and more remote. As the mir- rors are turned, approaching a right angle with each other, the number of repetitions grows less, and the whole take a circular arrange- ment. At a right angle, the object and three images are visible, arranged as represented in fig. 4. The rays O and O D from the point O, after a single reflection, give, the one an image O', and the other an image O", while the ray O A, which has undergone two reflec- tions at A and B, gives a third image O'". When the angle of the mirrors is 60, five images are produced, and seven when it is 45. The number of images continues to increase in proportion as the angle diminishes, and when it is zero, that is, 'when the mirrors are par- allel, the number of images is theoretically o -i o FIG. 4. infinite. (See KALEIDOSCOPE.) The amount of light reflected from a surface of given size and polish is different with mirrors of different material ; and it increases in all cases with in- crease of the angle of incidence, though not in all cases regularly. We observe the image of the sun in water near midday without difficul- ty ; but when near the horizon the brightness of the reflected light is usually intolerable. Eemembering that the surface impinged on by any single ray of light is extremely small, it will be seen that any curved reflector is in effect simply a collection of a great number of such minute planes ; and that, if we consid- er the rays falling on such a surface as reflect- ed from the same points in as many different planes tangent to the surface at the points of incidence, we at once extend the law for plane surfaces to all curved surfaces whatever. To the points of incidence of rays on any curved surface, K A B, fig. 5, let fall lines K, C I, A, &c., perpendicular (normal) to the surface at those points ; each reflected ray will be in the plane containing its incident ray and its proper normal ; and the angles of reflection, C K Z, 1 1, &c., and of incidence, L K C, L I 0, &c., will be