Page:The American Cyclopædia (1879) Volume XII.djvu/670

 656 OPTICS equal for each ray on the two sides of its nor- mal. Ordinary concave and convex mirrors are parts of spherical surfaces. The former must reflect parallel rays convergent, convergent rays more rapidly so, &c. The latter must re- flect parallel rays divergent, divergent rays more FIG. 5. so, &c. Parallel rays falling on a concave mir- ror are reflected to a focus distant from the surface half the radius of curvature of such surface, i. e., at one fourth the diameter of the sphere, as shown in fig. 6, where D being the normal at the point of incidence D, the angle -e- FIG 6. of reflection D F is equal to the angle of in- cidence GDC, and is in the same plane. It follows from this that the point F, where the reflected ray cuts the principal axis, divides the radius of curvature A very nearly into two equal parts. For in the triangle D F 0, the an- gle D F is equal to the angle D G, since they are alternate and opposite angles ; likewise the angle C D F is equal to the angle C D G, from the laws of reflection ; therefore the angle F D C is equal to the angle F C D, and the sides F C and F D are equal as being opposite to equal an- gles. The smaller the arc A D, the more near- ly does D F equal A F ; and when the arc is only a small number of degrees, the right lines A F and F may be taken as approximate- ly equal, and the point F may be taken as the middle of A 0. So long as the aperture of the mirror does not exceed 8 or 10, any other ray, H B, will after reflection pass very nearly through the point F. Hence, when a pencil of rays parallel to the axis falls on a concave mir- ror, the rays intersect after reflection in the same point, which is at an equal distance from the centre of curvature and from the mirror. This point is called the principal focus of the mir- ror, and the distance A F is the principal focal distance. If the angle of aperture of the mir- ror exceeds 10, not all of the reflected rays will meet in one and the same focal point, but, by reason of the various angles of incidence made by the incident rays on the curved sur- face, the further the point of incidence of a ray is from the centre M of the mirror A M B, fig. 7, the nearer to that centre will the ray be re- flected ; but incident rays included in an angle of aperture of 10 will approximately be re- flected to one focus F. Fig. 7 is an accurate representation of the paths of the reflected ray of an incident beam of parallel rays. M is the centre of figure of the spherical mirror A M B, C is the centre of curvature, and F is the focus. This departure from a true focus of rays re- flected from spherical mirrors is called "spher- ical aberration." The curved line A L F formed by the intersections of the reflected rays is called a " caustic." This caustic can be easily seen by placing a piece of paper in the same plane with the axis of the mirror, or by observ- ing the reflection from a curved polished clock spring placed on a piece of white paper in the sunshine. Spherical aberration can only be avoided by using mirrors of small angles of B FIG. 7. aperture, or by the use of mirrors having para- boloid surfaces, as shown in section in fig. 8. It is a well known property of the parabola that a normal bisects the angle made by a di- 3C FIG. 8. ameter at the point of contact with the line drawn from that point to the focus ; hence all rays, E M, O B, in fig. 8, parallel to the princi- pal axis A X, will be reflected to one point F, the focus of the mirror ; and conversely, if F b@ a luminous point, all rays emanating from