In any event, since the spherical aberration of a lens may be varied in above the ratio of 4:1, for the same total power, merely by changing the ratio of the radii, it is evident that the two lenses being fairly correct in total curvature might be given considerable variations in curvature and still mutually annul the axial spherical aberration. Such is in fact the case, so that to get determinate forms for the lenses one must introduce some further condition or make some assumption that will pin down the separate curvatures to some definite relations. The requirement may be entirely arbitrary, but in working out the theory of objectives has usually been chosen to give the lens some real or hypothetical additional advantage. The commonest arbitrary requirement is that the crown glass lens shall be equiconvex, merely to avoid making an extra tool. This fixes one pair of radii, and the flint lens is then given the required compensating aberration choosing the easiest form to make. This results in the objective of Fig. 51. [Illustration: FIG. 51.—Objectives with Equiconvex Crown.] Probably nine tenths of all objectives are of this general form, equiconvex crown and nearly or quite plano-concave flint. The inside radii may be the same, in which case the lenses should be cemented, or they may differ slightly in either direction as _a_, Fig. 51 with the front of the flint less curved than the rear of the crown, and _b_ where the flint has the sharper curve. The resulting lens if ordinary glasses are chosen gives excellent correction of the spherical aberration on the axis, but not much away from it, yielding a rather narrow sharp field. Only a few exceptional combinations of glasses relieve this situation materially. The identity of the inner radii so that the surfaces can be cemented is known historically as Clairault’s condition, and since it fixes two curvatures at identity somewhat limits the choice of glasses, while to get proper corrections demands quite wide variations in the contact radii for comparatively small variations in the optical constants of the glass. When two adjacent curves are identical they should be cemented, otherwise rays reflected from say the third surface of Fig. 51 will be reflected again from the second surface, and passing through the rear lens in almost the path of the original ray will come to nearly the same focus, producing a troublesome “ghost.” Hence the curvatures of the second and third surfaces when not cemented are varied one way or the other by two or three per cent, enough to throw the twice reflected rays far out of focus.