But obviously for large angles of incidence these convenient equalities do not hold. As the upper half of the figure shows, the ray parallel to the axis and incident on the mirror 45° away at _e_ is turned straight down, for it falls upon a surface inclined to it by 45° and is therefore deflected by 90°, cutting the axis far inside the nominal focus, at _d_. Following up other rays nearer the axis it appears that there is no longer a focal point but a cusp-like focal surface, known to geometrical optics as a caustic and permitting no well defined image. A paraboloidal reflecting surface as in Fig. 65 has the property of bringing to a single point focus all rays parallel to its axis while quite failing of uniting rays proceeding from any point on its axis, since its curvature is changing all the way out from vertex to periphery. Here the parallel rays _a_, _a_, _a_, _a_ meeting the surface are reflected to the focus _F_, while in a perfectly symmetrical way the prolongation of these rays _a′_, _a′_, _a′_, _a′_ if incident on the convex surface of the paraboloid would be reflected in _R_, _R′_, _R″_ _R″′_ just as if they proceeded from the same focus _F_. The difference between the spherical and parabolic curves is well shown in Fig. 66. Here are sections of the former, and in dotted lines of the latter. The difference points the moral. The parabola falls away toward the periphery and hence pushes outward the marginal rays. But it is of relatively sharper curvature near the center and pulls in the central to meet the marginal portion. In the actual construction of parabolic mirrors one always starts with a sphere which is easy to test for precision of figure at its center of curvature. Then the surface may be modified into a paraboloid lessening the curvature towards the periphery, or by increasing the curvature toward the center starting in this case with a sphere of a bit longer radius as in Fig. 66a. [Illustration: FIG. 65.—Reflection from Paraboloid.] [Illustration: FIG. 66_a_. FIG. 66_b_. Variation of Paraboloid from Sphere.] Practice differs in this respect, either process leading to the same result. In any case the departure from the spherical curve is very slight—a few hundred thousandths or at most ten thousandths of an inch depending on the size and relative focus of the mirror. Yet this small variation makes all the difference between admirable and hopelessly bad definition. However the work is done it is guided by frequent testing, until the performance shows that a truly parabolic figure has been reached. Its attainment is a matter of skilled judgment and experience.