LIGHT 587 leal and irtnal It is to be remarked that there are exceptions to this form of the statement. The true form is that the actual path of a ray, under the given conditions, is less in length than any other path (satis fying the conditions) which is nowhere finitely divergent from it. This may be best seen by another method. Suppose a series of ellipses to be described whose foci are the source of light and an assigned point which is to be reached by the reflected ray. Let this system turn about the line joining the foci; it generates a series of prolate spheroids such that the time of light s passing from one focus to a point in any one of the surfaces, and thence to the other focus, is the same whatever point be chosen on that particular sur face. If we take any point icithout that surface, for it the corre sponding time is obviously greater. Hence to find the path of that one of a system of rays diverging from a given point which, after reflexion at a given surface, shall pass through a given point, vre have only to imagine spheroids constructed as before. Of these one at least will touch the given surface. All points on the surface in the neighbourhood of the point of contact (mark this limitation) will iu general be outside the spheroid. Hence this point gives the shortest path. But the spheroid and the reflecting surface have the same tangent plane, and therefore the parts of the ray are equally inclined to the surface. Formation of Images by Reflexion at a Plane Surface. We may assume here what is indeed evident from the rectilinear propagation of light that objects are rendered visible to the eye by rays diverging from them. Hence, if we have a set of reflected or refracted rays diverging from any point, or diverging as if they came from any point, they will convey to the eye the impression of the existence of a luminous source at that point. The eye, in fact, can only tell us what effect is produced upon it, i.e., what sort of mechanical action it is subjected to. Its indications must therefore depend only upon what reaches it, and in no other sense whatever upon the source or the path of light. This point from which rays diverge, or appear to diverge, is called an image. The image of any point in a plane mirror is found by drawing from the point a perpendicular on the mirror and producing it till its length is doiibled. The extremity of the line so drawn is the image of the point ; or, in other words, rays proceeding from the point diverge after reflexion as if they came from the image so found. The image in this case is called virtual, to dis tinguish it from cases, subsequently to be mentioned, where it is real the distinction being that the rays have actually passed through a real image, while they only appear to come from a virtual one. To prove this it is only necessary to observe that, if A (fig. 5) be a luminous point or a point from which rays Fig. 5. diverge, and CB any section of the mirror by a plane through AB, the perpendicular to it, and if we make B = AB, and take any point P, then, joining AP, aP, and producing the latter, the angles APB, r/PB, and therefore CPQ are equal; also the plane of the piper contains the perpendicular to the mirror at P. Hence PQ is the reflected ray ; or the ray, after reflexion, appears Dircks s ghost. to come from a. Hence a is the image a virtual one, as before noticed. Also, if R be any point whatever (not P) in the plane of the mirror, we have obviously aR = RA. Hence the path All, EQ is equal to aB, BQ, two sides of the triangle aRQ, of which aQ, which is equal to the actual path (AP, PQ), is the third side. Fig. 6 represents the pencils of diverging rays by which two points of the image are rendered visible to an eye placed in front of the mirror. From the re- 1 quisite modifi cation of this figure it follows that one can seel his whole per son in a mirror of only half his height and breadth. Dircks s ghost, which has played a prominent part in popular enter tainments for some years back, is the image, in a large sheet of unsilvered plate glass hung at the front of the stage, of an actor or figure strongly illuminated, and concealed from the audience in a sort of enlarged prompter s box. Any one can see the phenomenon completely by looking into a plate-glass win dow on a sunny day, when he sees the passers-by ap parently walking inside the house. The principles already stated suffice fully for the ex planation of the curious vistas of images formed by two parallel plane mirrors facing one another at opposite sides of a room. The only additional observation necessary on this subject is that, if the mirrors are silvered on the back, the light at each reflexion has to pass twice through the glass. Thus, if the glass be pinkish or greenish, the various images are more and more coloured as they are due to more numerous reflexions. These principles also easily explain the KALEIDOSCOPE Kaleido- (q.v.) of Sir D. Brewster, where images are formed by two sc P e - mirrors inclined to one another. It is easy to see that the series of images of a luminous point produced by such an arrangement after one, two, &c., reflexions must all lie on a circle ; also that, if the angle between the mirrors be an aliquot part of four right angles, these images will form a finite number of groups, each consisting of an infinite number of images which have exactly the same position. The explanation of the law of reflexion which is fur nished by the corpuscular theory is excessively simple. We have only to suppose that at the instant of its impact on the reflecting surface the velocity of a corpuscle per pendicular to the surface is reversed, while that parallel to the surface is unchanged. It bounds off. in fact, like a billiard-ball from the cushion. The undulatory theory gives an explanation, which is s in reality, quite as simple, but requires a little more detail for those who are not familiar with the common facts of wave-motion. We therefore reserve it for a time. Reflexion at a Spherical Surface. Let APB (fig. 7) be Spherical a section of a concave spherical mirror by a plane passing mirror. . through its centre of curvature O, and through the lumi nous point U. Then, if any ray from U, as UP, meet the surface, it will be reflected in a direction PV, such that UP, PO, and PY are in one plane, and so tha f PO bisects
