Such a beam _a_, _a_, _a_, in fact behaves as if it came from and returned to a virtual conjugate focus _F′_ on the other side of the hyperbolic surface. And if the convex side be reflecting, converging rays _R_, _R_′, _R″_, falling upon it at _P_, _P′_, _P″_, as if headed for the virtual focus _F_, will actually be reflected to _F′_, now a real focus. This surface being convex its aberrations off the axis are of opposite sign to those due to a concave surface, and can in part at least, be made to compensate the aberrations of a parabolic main mirror. The rationale of the operation appears from comparison of Figs. 67 and 68. [Illustration: FIG. 68.—Reflection from Hyperboloid.] In the former the oblique rays _a_, _a′_ are pitched too sharply down. When reflected from the convex surface of Fig. 68 as a converging beam along _R_, _R′_, _R″_, they can nevertheless, if the hyperbola be properly proportioned, be brought down to focus at _F′_ conjugate to _F_, their approximate mutual point of convergence. Evidently, however, this compensation cannot be complete over a wide angle, when _F′_ spreads into a surface, and the net result is that while the total aberrations are materially reduced there is some residual coma together with some increase of curvature of field, and distortion. Here just as in the parabolizing of the large speculum the construction is substantially empirical, guided in the case of a skilled operator by a sort of insight derived from experience. Starting from a substantially spherical convexity of very nearly the required curvature the figure is gradually modified as in the earlier example until test with the truly parabolic mirror shows a flawless image for the combination. The truth is that no conic surface of revolution save the sphere can be ground to true figure by any rigorous geometrical method. The result must depend on the skill with which one by machine or hand can gauge minute departures from the sphere. Attempts have been made by the late Professor Schwarzchild and others to improve the corrections of reflectors so as to increase the field but they demand either very difficult curvatures imposed on both mirrors, or the interposition of lenses, and have thus far reached no practical result. REFERENCES SCHWARZCHILD: Untersuchungen 2, Geom., Opt. II. SAMPSON _Observatory 36_, 248. CODDINGTON: “Reflexion and Refraction of Light.” HERSCHEL: “Light.” TAYLOR: “Applied Optics.” SOUTHALL: “Geometrical Optics.” MARTIN: _Ann. Sci. de l’Ecole Normale_, 1877, Supplement. MOSER: _Zeit. f._ Instrumentenkunde, 1887. HARTING: _Zeit. f. Inst._, 1899. HARTING: _Zeit. f. Inst._, 1898. VON HOEGH: _Zeit. f. Inst._, 1899. STEINHEIL & VOIT: “Applied Optics.” COLLECTED RESEARCHES, National Physical Laboratory, Vol. 14. GLEICHEN: “Lehrbuch d. Geometrische Optik.”