SPHERICAL ABERRATION. 53 



nection with others, which will facilitate its comprehension. 

 Here we need only remark that the blue fringe encircling the 

 field of view, or on the outline of a rather large object, does not 

 prove that the objective is really over-corrected, and still less 

 that it possesses a chromatic preponderance in proportion to the 

 under-corrected lenses of the eye-piece. 



2. SPHERICAL ABERRATION. 



Spherical aberration, or aberration caused by the spherical form, 

 is due to the fact that the focal length of the marginal rays is 

 always less than that of the central rays, and diminishes with 

 the distance from the axis. If the surface of the lens be divided 

 into concentric zones, each zone represents a different focal length, 

 and therefore also a different position of the image. The com- 

 bination of flint and crown serves also to correct these defects. 

 It is possible to establish such a ratio of the refractions in an 

 achromatic double-lens that the marginal rays, which are de- 

 flected from the axis by the concave flint to a greater extent 

 than the central rays, shall be more strongly refracted by the 

 crown in an opposite direction, their focus then coinciding with 

 that of the central rays. A double-lens in which all the rays 

 are brought to a focus in one plane is called aplanatic. 1 



Every achromatic pair of lenses can be made aplanatic also. 

 For since achromatism depends, as is apparent from the equation 

 given at p. 51, upon the focal lengths and dispersive ratios, and 

 aplanatism, 011 the other hand, upon the curvatures, the two 

 conditions are not antagonistic. If the anterior surface of the 

 flint-lens is given, the radii of the other surfaces can be com- 

 puted so that all aberration will disappear. If, for instance, a 

 double-lens with a plane anterior surface and equal curvatures 

 is achromatic but not aplanatic, that is, if the influence of the 

 flint upon the marginal rays is too weak, all that is necessary 

 is a slight increase in the curvature of the surfaces of contact 

 of the two lenses, and a corresponding alteration in that of the 

 posterior surface of the crown, the ratio between the focal lengths 



1 Many authors give a somewhat more extended meaning to this expression. 

 They call a system of lenses aplanatic, when not only spherical but also chro- 

 matic aberration is eliminated as far as possible. We follow Radicke (" Hand- 

 buch der Optik") and other mathematicians on this point. 



