Even the diffraction theory can be taken only as an approximation since no optical surface is absolutely perfect and in the ordinary refracting telescope there is a necessary residual chromatic aberration beside whatever may remain of spherical errors. [Illustration: FIG. 186.—Light-grasp and Resolving Power.] It is a fact therefore, as has been shown by Conrady (M.N. =79= 575) following up a distinguished investigation by Lord Rayleigh (Sci. Papers =1= 415), that a certain small amount of aberration can be tolerated without material effect on the definition, which is very fortunate considering that the secondary spectrum represents aberrations of about 1/2,000 of the focal length, as we have already seen. The chief effect of this, as of very slight spherical aberration, is merely to reduce the maximum intensity of the central disc of the diffraction pattern and to produce a faint haze about it which slightly illuminates the diffraction minima. The visible diameter of the disc and the relative distribution of intensity in it is not however materially changed so that the main effect is a little loss and scattering of light. With larger aberrations these effects are more serious but where the change in length of optical path between the ray proceeding through the center of the objective and that from the margin does not exceed ¼λ the injury to the definition is substantially negligible and virtually disappears when the image is focussed for the best definition, the loss of maximum intensity in the star disc amounting to less than 20%. Even twice this error is not a very serious matter and can be for the most part compensated by a minute change of focus as is very beautifully shown in a paper by Buxton(M. N. =81=, 547), which should be consulted for detail of the variations to be effected. Conrady finds a given change _dp_ in the difference in lengths of the optical paths, related to the equivalent linear change of focus, _df_, as follows:— _df_ = 8_dp_(_f_/_A_)² A being the aperture and f the focal length, which indicates for telescopes of ordinary focal ratio a tolerance of the order of ±0.01 inch before getting outside the limit λ for variation of path. For instruments of greater relative aperture the precision of focus and in general the requirements for lessened aberration are far more severe, proportional in fact to the square of this aperture. Hence the severe demands on a reflector for exact figure. An instrument working at F/5 or F/6 is extremely sensitive to focus and demands great precision of figure to fall within permissible values, say ¼λ to ½λ, for _dp_. Further, with a given value of _dp_ and the relation established by the chromatic aberration, _i.e._, about _f_/2000, a relation is also determined between _f_ and _A_, required to bring the aberration within limits. The equation thus found is _f_ = 2.8_A_²