The determination of achromatism for any pair of glasses and focal length is greatly facilitated by employing the auxiliary quantity ν which is tabulated in all lists of optical glass as a short cut to a somewhat less manageable algebraic expression. Using this we can figure achromatism for unity focal length at once, P = ν/(ν-ν′) P′ = ν′/(ν-ν′) ν = (n_{_D_}-1)/δn being the powers of the leading and following lenses respectively. The combined lens will bring the rays of the two chosen colors, as red and blue, to focus at the same point on the axis. It does not necessarily give to the red and blue images of an object the same exact size. Failure in this respect is known as chromatic difference of magnification, but the fault is small and may generally be neglected in telescope objectives. We have now seen how an objective may be made achromatic and of determinate focal length, but the solution is in terms of the sums of the respective curvatures of the crown and flint lenses, and gives no information about the radii of the individual surfaces. The relation between these is all-important in the final performance. [Illustration: FIG. 49.—Spherical Aberration of Convex Lens.] For in a convex lens with spherical surfaces the rays striking near the edge, of whatever color, are pitched inwards too much compared with rays striking the more moderate curvatures near the axis, as shown in Fig. 49. The ray _a′ b′_ thus comes to a focus shorter than the ray _a b_. This constitutes the fault of spherical aberration, which the old astronomers, following the suggestions of Descartes, tried ineffectually to cure by forming lenses with non-spherical surfaces. [Illustration: FIG. 50.—Spherical Aberration of Concave Lens.] Fig. 50 suggests the remedy, for the outer ray _a″_ is pitched out toward _b″_ as if it came from a focal point _c″_, while the ray nearer the center _a″′_ is much less bent toward _b″′_ as if it came from _c″′_. The spherical aberrations of a concave lens therefore, being opposite to those of a convex lens, the two must, at least to a certain extent, compensate each other as when combined in an achromatic objective. So in fact they do, and, if the curves that go to make up the total curvatures of the two are properly chosen, the total spherical aberration can be made negligibly small, at least on and near the axis. Taking into account this condition, therefore, at once gives us a clue to the distribution of the total curvatures and hence to the radii of the two lenses. Spherical aberration, however, involves not only the curvatures but the indices of refraction, so that exact correction depends in part on the choice of glasses wherewith to obtain achromatization. In amount spherical aberration varies with the square of the aperture and inversely with the cube of the focal length i.e. with a²/f³. It is reckoned as + when, as in Fig. 49, the rim rays come to the shorter focus, as-, when they come to the longer focus.