In this case, as in most others, the analytical expression for the fundamental curvature to be determined turns up in the form of a quadratic equation, so that the result takes the form a ± b and there are two sets of radii that meet the requirements. Of these the one presenting the gentler curves is ordinarily chosen. Fig. 52 _a_ and _c_ shows the two cemented forms, thus related, for a common pair of crown and flint glasses, both cleanly corrected for chromatic and axial spherical aberration. Nearly a century ago Sir John Herschel proposed another defining condition, that the spherical aberration should be removed both for parallel incident rays and for those proceeding from a nearer point on the axis, say ten or more times the focal length in front of the objective. This condition had little practical value in itself, and its chief merit was that it approximated one that became of real importance if the second point were taken far enough away. [Illustration: FIG. 52.—Allied Forms of Cemented Objectives.] A little later Gauss suggested that the spherical aberration should be annulled for two different colors, much as the chromatic aberration is treated. And, being a mathematical wizard, he succeeded in working out the very intricate theory, which resulted in an objective approximately of the form shown in Fig. 53. It does not give a wide field but is valuable for spectroscopic work, where keen definition in all colors is essential. Troublesome to compute, and difficult to mount and center, the type has not been much used, though there are fine examples of about 9½ inches aperture at Princeton, Utrecht, and Copenhagen, and a few smaller ones elsewhere, chiefly for spectroscopic use. It was Fraunhofer who found and applied the determining condition of the highest practical value for most purposes. This condition was absence of _coma_, the comet shaped blur generally seen in the outer portions of a wide field. It is due to the fact that parallel oblique rays passing through the opposite rims of the lens and through points near its center do not commonly come to the same focus, and it practically is akin to a spherical aberration for oblique rays which greatly reduces the extent of the sharp field. It is reckoned + when the blur points outwards,-when it points inwards, and is directly proportional to the tangent of the obliquity and the square of the aperture, and inversely to the square of the focal length i.e. it varies with a²tan(u)/f². [Illustration: FIG. 53.—Gaussian Objective.]