54 



THEORY OF THE MICROSCOPE. 



remaining therefore unaffected. In the opposite case, the radius 

 of the surfaces of contact must be increased, and then the 

 posterior surface of the crown must be determined from the 

 equation for achromatism. For the practical optician it is most 

 advantageous that the different kinds of glass should be so 

 constituted as to allow the radii of the crown to be almost 

 equal, the shortest focal lengths being obtained by such com- 

 binations with the strongest practicable curvatures. 



In the strictest sense of the word, aplanatism is just as little 

 attainable as achromatism. Marginal rays are acted upon by 

 Hint-lenses in proportion to their distance from the axis. If, 

 therefore, the focus of the extreme marginal rays coincides with 

 that of the central, the rays passing nearer to the axis will 

 always show traces of aberration. 



Moreover, calculation as well as experiment 

 shows that a double-lens, which is sufficiently 

 aplanatic for rays incident in a parallel direc- 

 tion, ceases to be so when the rays converge 

 fCC or diverge ; and that, in general, aplanatism 



^" can only be obtained for definite distances 

 of the (real or virtual) object-points. If, for 

 instance, the double-lens A B (Fig. 18) is 

 aplanatic for the point a lying somewhat 

 beyond the focus, then, if the object be brought 

 nearer, an over-correction is observed, which 

 gradually increases and then decreases until 

 it almost vanishes when the point & is reached. 

 With still greater proximity of the object, or 

 on its withdrawal beyond a, aberration in an 

 opposite direction, that is, under-correction, 

 a appears. 



FIG. 18. The two points a and &, the rays from which 



pass through without aberration, are called 

 the aplanatic foci of the double-lens ; at the same time it must 

 be observed that this term is not in all respects applicable, as 

 the aplanatism in question extends only to a zone of limited 

 dimensions, although of importance for the formation of the image. 

 This is especially the case with lenses of deep curvature. The 

 importance of these aplanatic foci has, therefore, been much 

 over-estimated in recent times, the residual aberration which is 



