This practically amounts to the common F/15 ratio for an aperture of approximately 5 inches. For smaller apertures a greater ratio can be well used, for larger, a relatively longer focus is indicated, the penalty being light spread into a halo over the diffraction image and reducing faint contrasts somewhat seriously. This is one of the factors aside from atmosphere, interfering with the full advantage of large apertures in refractors. While as already noted small amounts of spherical aberration may be to a certain extent focussed out, the sign of _df_ must change with the sign of the residual aberration, and a quick and certain test of the presence of spherical aberration is a variation in the appearance of the image inside and outside focus. To emphasize the importance of exact knowledge of existing aberrations note Fig. 187, which shows the results of Hartmann tests on a typical group of the world’s large objectives. All show traces of residual zones, but differing greatly in magnitude and position as the attached scales show. The most conspicuous aberrations are in the big Potsdam photographic refractor, the least are in the 24 inch Lowell refractor. The former has since been refigured by Schmidt and revised data are not yet available; the latter received its final figure from the Lundins after the last of the Clarks had passed on. Now a glance at the curves shows that the bad zone of the Potsdam glass was originally near the periphery, (I), hence both involved large area and, from Conrady’s equation, seriously enlarged _df_ due to the large relative aperture at the zone. An aberrant zone near the axis as in the stage (III) of the Potsdam objective or in the Ottawa 15 inch objective is much less harmful for corresponding reasons. Such differences have a direct bearing on the use of stops, since these may do good in case of peripheral aberration and harm when the faults are axial. Unless the aberrations are known no general conclusions can be drawn as to the effect of stops. Even in the Lowell telescope shown as a whole in Fig. 188, the late Dr. Lowell found stops to be useful in keeping down atmospheric troubles and reducing the illumination although they could have had no effect in relation to figure. Fig. 188 shows at the head of the tube a fitting for a big iris diaphragm, controlled from the eye-end, the value of which was well demonstrated by numerous observers. There are, too, cases in which a small instrument, despite intrinsic lack of resolving power, may actually do better work than a big one. Such are met in instances where extreme contrast of details is sought, as has been well pointed out by Nutting (Ap. J. =40=, 33) and the situation disclosed by him finds amplification in the extraordinary work done by Barnard with a cheap lantern lens of 1½ inch diameter and 5½ inches focus (Pop. Ast., =6=, 452). [Illustration: FIG. 187.—Hartmann Tests of Telescopes [From Hartmann’s Measures].]