Just how Fraunhofer solved the problem is quite unknown, but solve it he did, and very completely, as he indicates in one of his later papers in which he speaks of his objective as reducing all the aberrations to a minimum, and as Seidel proved 30 years later in the analysis of one of Fraunhofer’s objectives. Very probably he worked by tracing axial and oblique rays through the objective form by trigonometrical computation, thus finding his way to a standard form for the glasses he used.[10] [10] More recently his condition proves to be quite the exact equivalent of Abbé’s _sine condition_ which states that the sine of the angle made with the optical axis by a ray entering the objective from a given axial point shall bear a uniform ratio to the sine of the corresponding angle of emergence, whatever the point of incidence. For parallel rays along the axis this reduces to the requirement that the sines of the angles of emergence shall be proportional to the respective distances of the incident rays from the axis. Fraunhofer’s objective, of which Fig. 54_a_ is an example worked by modern formulæ for the sine condition, gives very exact corrections over a field of 2°-3° when the glasses are suitably chosen and hence is invaluable for any work requiring a wide angle of view. With certain combinations of glasses the coma-free condition may be combined successfully with Clairault’s, although ordinarily the coma-free form falls between the two forms clear of spherical aberration, as in Fig. 52, _b_, which has its oblique rays well compensated but retains serious axial faults. [Illustration: FIG. 54.—The Fraunhofer Types.] Fraunhofer’s objective has for all advantageous combinations of glasses the front radius of the flint longer than the rear radius of the crown hence the two must be separated by spacers at the edge, which in small lenses in simple cells is slightly inconvenient. However, the common attempt to simplify mounting by making the front flint radius the shorter almost invariably violates the sine condition and reduces the sharp field, fortunately not a very serious matter for most astronomical work. The only material objection to the Fraunhofer type is the strong curvature of the rear radius of the crown which gives a form somewhat susceptible to flexure in large objectives. This is met in the flint-ahead form, developed essentially by Steinheil, and used in most of the objectives of his famous firm. Fig. 54_b_ shows the flint-ahead objective corresponding to Fig. 54_a_. Obviously its curves are mechanically rather resistant to flexure.[11] [11] It is interesting to note that in computing Fig. 54_a_ for the sine condition, the other root of the quadratic gave roughly the Gaussian form of Fig. 53. [Illustration: FIG. 55.—Clark Objective.]