Page:Elementary Text-book of Physics (Anthony, 1897).djvu/434

420 is a property of an ellipse with foci at $$L$$ and $$l.$$ If the ellipse be constructed and revolved about $$Ll$$ as an axis, it will generate a surface which will have the required property. If one of the points $$L$$ be removed to an infinite distance, the corresponding wave becomes a plane perpendicular to $$Ll,$$ and we must have $$LB + BC$$ (Fig. 111) constant, a property of the parabola. A parabolic mirror will therefore concentrate at its focus incident light moving in paths parallel to its axis, or will reflect incident light diverging from its focus in plane waves perpendicular to its axis.

Mirrors and lenses having surfaces which are not spherical are seldom made because of mechanical difficulties of construction. It becomes necessary, therefore, to consider how the disadvantages arising from the use of spherical surfaces of large aperture for reflecting or refracting light may be avoided or reduced.

We will consider first the case of a spherical mirror. It was shown above that light from one focus of an ellipsoid is reflected from the ellipsoidal surface in perfectly spherical waves concentric with the other focus. Let Fig. 112 represent a plane section through the axis of an ellipsoid, and Fca a small incident pencil of light proceeding from the focus $$F.$$ $$F'ac$$ is a section of the reflected pencil. It is a property of the ellipse that the normals to the curve bisect the angles formed by lines to the two foci. The normal $$ae$$ bisects the angle $$FaF'$$ and hence in the triangle $$FaF'$$ we have $$\frac{Fa}{F'a} = \frac{Fe}{F'e}\cdot$$

If $$d$$ move toward $$c,$$ $$F'a$$ increases and $$Fa$$ diminishes. Hence, from the above proportion, $$F'e$$ must increase and $$Fe$$ diminish; or, the successive normals as we approach the minor axis cut the major axis in points successively nearer the centre of the ellipse. The normals produced will therefore meet each other at $$$$n beyond the axis. If $$ac$$ be taken small enough, it may be considered the arc of a circle of which $$an, cn$$ are radii and $$n$$ the centre. It is