When the opaque screen is at a given distance from the mirror under examination, the only parts of the mirror which can officiate well are those which have a curvature corresponding to a radius equal to the same distance. All the other parts seem as if they were covered by projecting circular masses. In looking at Fig. 14, it is plain, then, if the opaque screen is at a maximum distance from the mirror, that the central parts alone will seem to operate, because the two curves (_a_) only touch there. If the screen is moved toward the mirror the curves (_b_) will coincide at some point between the centre and edge, while if carried still farther in only the edges touch and the appearance will be as if a large mound were fixed upon the centre. I have been careful in explaining how a surface may thus seem to present entirely different characteristics if examined from points of view which vary slightly in distance, because a knowledge of these facts is of the utmost importance in correcting such an erroneous figure. It is now obvious that the correction will be equally effectual if the mirror be polished with a small rubber on the edge, or on the centre, or partly on each. The only difference in the result will be, that the mean focal length will be increased in the first instance, and decreased in the second, while it will remain unchanged in the third. [Illustration: Fig. 15. Caustic of Hyperbolic Mirror.] [Illustration: Fig. 16. Apparent Section of Hyperbolic Mirror.] If the mirror, instead of having a section like that of an oblate spheroid, should have either an ellipse, parabola, or hyperbola, as its section curve, the appearances seen above are reversed. Whilst by the first test there is still an aberration round the image at the best focus, the eye-piece must now be drawn from the mirror to include it. The cone of rays is most dense round the axis inside, and at the periphery outside the focus, and the summit of the caustic (Fig. 15) is turned towards the mirror. The second test shows a section as in Fig. 16, a depression at the centre, and the edges turned backwards. The nature of the movement necessary to reduce the surface to a sphere is very plainly indicated, action on a zone _a_ between the centre and edge. If, however, a parabolic section is required, the zone _a_ must not be entirely removed, and the surface rendered apparently flat, but as much of it must be left as experience shows to be desirable. [Illustration: Fig. 17. Action of the Opaque Screen.] [Illustration: Fig. 18. Apparent Section of Mirror with Rings.]