SPHERICAL ABERRATION. 55 



present having been regarded as the effect of a prevailing under- 

 or over-correction (i.e., extending over the whole extent of the 

 lens), and this idea has heen transferred without limitation to 

 the case of high-power objectives with large angles of aperture. 

 Calculation, however, shows that in this case the ratio of the 

 aberrations is a much more complicated one. " Spherical aberra- 

 tion may, on a strict investigation of its conditions, be divided 

 in general into a series of independent factors, which, as they 

 increase with the greater inclination of the rays towards the axis, 

 pursue a very unequal course. Complete elimination is theoreti- 

 cally possible only for the two first factors. As soon as the angle 

 of aperture exceeds a very small amount, the correction of spherical 

 aberration can only be attained by compensating the ineffaceable 

 higher terms by intentionally introduced residua of the lower 

 ones. The increase of the unavoidable deficit, which this com- 

 pensation necessarily leaves unremedied on account of the dis- 

 similar course of the separate parts, determines the limit which 

 must be set to the angle of aperture if this deficit is to remain 

 without injurious effect in the microscopic image." 1 



In the manufacture of objectives with large angles of aperture 

 (e.g., 60 to 80) the optician is therefore referred chiefly to 

 his own experience; he must discover by repeated experiment 

 not only the most suitable kinds of glass, but also the correct 

 ratios of the radii of curvature and of the distances of the lenses, 

 and must endeavour, in this purely empirical way, to eliminate, 

 as far as possible, both the aberrations. It is clear that a 

 preliminary theoretical testing of different combinations will 

 always afford many valuable hints ; for instance, it is an in- 

 dispensable condition in high-power modern objectives with 

 angular apertures of 100, and upwards, that a single, nearly 

 hemispherical, front-lens should be combined with a system of 

 strongly over-corrected lenses. The discovery of this type of 

 construction must be regarded as the true foundation of all 

 recent improvements in objectives. 



Whatever may be the path of the rays for the peripheral parts 

 of the objective, it can always be so far improved by the alteration 

 of a lens-distance (as, for instance, by moving the correction- 

 collar) that at least the extreme marginal zone and the central 

 part of the objective will work together. But the intermediate 

 1 Abbe : " Archiv fur mikr. Anat." Bd. ix. p. 425. 



