The weak point of the parabolic mirror is in dealing with rays coming in parallel but oblique to the axis. Figure 67 shows the situation plainly enough. The reflected rays _a′_, _a″_ no longer meet in a point at the focus _F_ but inside the focus for parallel rays, at _f_ forming a surface of aberration. The practical effect is that the image rapidly deteriorates as the star passes away from the axis, taking on an oval character that suggests a bad case of astigmatism with serious complications from coma, which in fact is substantially the case. [Illustration: FIG. 67.—Aberration of Parabolic Mirror.] Even when the angular aperture is very small the focal surface is nevertheless a sphere of radius equal to one half the focal length, and the aberrations off the axis increase approximately as the square of the relative aperture, and directly as the angular distance from the axis. The even tolerably sharp field of the mirror is therefore generally small, rarely over 30′ of arc as mirrors are customarily proportioned. At the relative aperture usual with refractors, say F/15, the sharp fields of the two are quite comparable in extent. The most effective help for the usual aberrations[14] of the mirror is the adoption of the Cassegrain form, by all odds the most convenient for large instruments, with a hyperboloid secondary mirror. [14] A very useful treatment of the aberrations of parabolic mirrors by Poor is in Ap. J. 7, 114. In this is given a table of the maximum dimension of a star disc off the axis in reflectors of various apertures. This table condenses to the closely approximate formula a = lld/f² where a is the aberrational diameter of the star disc, in seconds of arc, d the distance from the axis in minutes of arc, f the denominator of the F ratio (F/8 &c.) and 11, a constant. Obviously the separating power of a telescope (see Chap. X) being substantially 4.″56/D where D is the diameter of objective or mirror in inches, the separating power will be impaired when a > 4.″56/D. In the photographic case the critical quantity is not 4.″56/D, but the maximum image diameter tolerable for the purpose in hand. mirror is the adoption of the Cassegrain form, by all odds the most convenient for large instruments, with a hyperboloidal secondary mirror. The hyperboloid is a curve of very interesting optical properties. Just as a spherical mirror returns again rays proceeding from its center of curvature without aberration, and the paraboloid sends from its focus a parallel axial beam free of aberration, or returns such a beam to an exact focus again, so a hyperboloid, Fig. 68, sends out a divergent beam free from aberration or brings it, returning, to an exact focus.