LIGHT 539 of the centre of curvature with the object ; inverted, if 0:1 the opposite. In other words, the image is inverted it the rays cross one another s path, erect if they do not. When the breadth of the mirror is large compared with its radius, the approximation upon which all these results depend can no longer bs made. There is then no definite image even of a luminous point. It becomes spread over what is called a caustic, a section of which is the bright curve familiar to every one who has looked at a cup of milk in sunshine. Even when the approximation 13 close enough for ordinary purposes, it is not so for astronomical purposes, and the effect of its inexactness upon the image is known as spherical aberration. For the fins mirrors of reflecting TELESCOPES (q.v.) the spherical form cannot be employed; the surface of the mirror must be of parabolic section. Cylindrical Mirrors. As a simple example of the ap plication of the law of reflexion at curved surfaces, when the rigorous solution is demanded, let us take the case of a vertical right cylinder, the object being a drawing on a horizontal plane. Such mirrors, with the frightfully dis torted drawings necessary to give an image of natural proportions, were very common fifty years ago, but are now rarely seen. They are still, however, valuable as illustrations of our subject. Let the plane of the object cut the axis OB of the cylinder at right angles in (fig. 10), and let A be the position of the eye, and RQA a ray from a point 11 of the ob ject, reflected at Q. Draw QP perpendicu lar to the axis. Then AQ and QR are in the same plane with QP (the normal to the surface) and make equal angles with it. Hence, when this figure is projected by vertical lines on the plane of the object, it takes the form in fig. 11; and AQ, QR now makeequalangleswith OQ. Also, if AB be drawn (in lig. 10) per pendicular to OP, tlis Fig. 10. ratio of AQ to QR in fig. ] 1 is equal to that of BP to PO in fig. 10. Diagrams fov . cvlm ~ dl . ical mirror. Reflexion from Fig. 11. Take QS : QO : : QR : QA, and draw ST parallel to OA. Then it is obvious that SR-ST-||OA; and also that the angles QSR and QST are equal. Hence the following theorems, which enable us at once to draw a figure on the object plane such that its image shall appear of any assigned form. 1. Any line, such as QR, on the object plane, drawn from a point Q iu the section of the cylinder so that the angles OQR and OQA are equal, is seen after reflexion as a generating line of the cylinder. 2. If an epicycloid be described by lines of fixed length OS, SR, turning about with angular velocities 1 and 2, and both coin ciding with OA at starting, its image will be a circular section of the cylinder. Thus, if we imagine as drawn on the cylinder any number of vertical and horizontal sections, forming a network, the object corresponding to them can be traced as a number c i. L- i i. T i i i mi ot intersecting straight lines and epicycloids. Thus we have a well-known means of drawing the required object. A similar process may be applied to other modes of using such mirrors. When the cylinder has a small diameter, it may be usefully employed to intercept and reflect part of a beam of sunlight entering a dark room. It is easy to see, by a geometrical construction, that the reflected rays will, in this case, form a right cone, whose axis is that of the cylinder ; and one of its generating lines will be parallel to the incident ray. Thus the angle of the cone becomes smaller as the inclination of the reflecting cylinder to the ray becomes less. If the ray, at the point of interruption, was at the centre of a spherical dome, after reflexion it will form on the dome a circle, small or great, which passes through its original point of incidence. In the language of QUATERNIONS (q.v.), let a be the incident ray, $ the axis of the cylinder, r any normal to the cylinder, p the reflected ray. Then the law of reflexion gives V.rarp=-0. The property of the normal gives S0T = 0. Eliminating r, we have at once the equation of a right cone. Imitations, more or less perfect, of primary and second- iry rainbows can easily be made by this process, the sun beam being led through a prism just before it falls on the cylindrical rod. This experiment is a very striking one ; but, though capable of giving much information, it is of that dangerous kind which is liable to mislead instead of instructing an audience. If we look at a great number of thin cylindrical rods, parallel to one another, and illuminated by sunlight, the rays which reach the eye must, by what we have already- said, each form a side of some right cone (of definite angle) whose axis is parallel to each of the cylinders. The appearance presented will therefore be that of a luminous circle, passing through the sun. Its angular diameter becomes less as the axes of the cylinders are less inclined to the incident rays. This phenomenon is beautifully shown by some specimens of crystals, especially of Iceland spar, which are full of minute tubes parallel to one another. In a plate of such a doubly-refracting crystal, however, there are necessarily four images. That which is throughout due to the ordinary ray (this term will be explained later) shows perfectly the phenomenon above described. The light of the luminous circle is white. The other three curves are not circles, and in them the colours are separated. One of them, which is elliptical, is usually very much brighter than either of the remaining two. REFRACTION. If homogeneous light be refracted at a Onlinary plane surface separating two homogeneous isotropic media, ref r:ic - the incident and refracted rays are in one plane uith, the tl0 " normal to the surface, and the sines of their inclinations to it are in a constant ratio.
