RATIO OF APERTURE AND FOCAL LENGTH. 99 



side, the whole middle portion of the aperture is therefore in- 

 operative. The objective may therefore be free from aberration 

 in the narrow effective circumferential zone, a condition which is 

 easily attainable by mere alteration of the distances of the lenses, 

 even with constructions in other respects defective, and the two 

 images coincide ; the action of the objective will therefore, in this 

 special case, be satisfactory, or possibly very favourable. It only 

 implies, however, that the angular aperture is large enough to admit 

 at least two of the pencils contributing to the formation of the 

 image, one diffracted, and one undiffracted, and that it is possible 

 to correct a narrow peripheral zone with sufficient accuracy. The 

 observer thus determines merely the limit, not the power, of differ- 

 entiation for cases which ordinarily occur. 



The formation of the image with axial illumination takes place 

 similarly. In some cases the direct rays pass through the middle 

 and also very differently situated parts of the objective ; but the 

 diffracted pencils, which reproduce the details of the object at the 

 limit of differentiating power, touch the margin of the free aperture, 

 and produce, therefore, a well-defined diffraction image, so far as 

 the marginal zone is accurately corrected. If, then, the dioptric 

 image is very indistinct, and the coincidence of the two images 

 very inexact, the delineation of fine structures remains, of course, 

 unaffected, and the superficial student, who attaches undue impor- 

 tance to this point, would estimate the quality of the objective too 

 highly. 



From these special cases we again arrive at the result, that an 

 accurate fusion of the interference with the dioptric images, com- 

 bined with a sufficient sharpness of delineation, is only possible if 

 the objective is uniformly aplanatic throughout its aperture. 



4. RATIO OF APERTURE AND FOCAL LENGTH. ^ 



The consideration of the fact that the magnifying power of a 

 Microscope is dependent on its focal length, whereas the diameter 

 of the details just perceptible in the microscopic image is solely 

 dependent on the angle of aperture, leads necessarily to the con- 

 clusion that both these factors, equally indispensable for the 

 delineation of small objects, must bear a fixed arithmetical ratio if 



H 2 



