The Belation of Ajperture and Power. By Prof. E. Abbe. 799 



quantity u would be = 0. In that case the dissipation of the 

 light at the ultimate image of the Microscope would also = 0, i. e. 

 the image would retain its full ideal perfection, with every amount 

 of super-amplification; and it would, therefore, be entirely un- 

 important whether a certain total power had been obtained by an 

 objective of long or short focal length. If, however, the objectives 

 in question are afflicted with certain defects, however small, the 

 quantity u will obtain a certain value ; and the dissipation of the 

 rays corresponding to that value, being more and more enlarged as 

 the super-amplification is increased, there must always exist a 

 certain maximum super-amplification or value of v which the objec- 

 tive will bear without a visible or an objectionable loss of sharpness 

 or perfection of image. Consequently we have a maximum of the 

 total amplification N which can be obtained with a given focal length 

 under the condition of a certain degree of sharpness of the image ; 

 and, conversely, a maximum focal length which admits of a given 

 amplification under the same conditions. If, for example, the 

 inherent dissipation of a certain kind of objective were confined to 

 an angle of 15", no eye would recognize the dissipation-circles if 

 such a system were used only under its own normal amplification 

 (using it as a simple Microscope). The dissipation would, how- 

 ever, become visible, and would introduce a perceptible indistinctness 

 of the image, if the super-amplification much exceeded 4, and the 

 deterioration would become very great should it amount to 16, because 

 in these cases the circles of indistinctness would be displayed in the 

 respective ultimate images under a visual angle of more than 1' 

 and of 4'. If now a certain amplification, say 320 diameters, is 

 required with objectives of that degree of perfection, a l-8th inch 

 would yield that number with a super-amplification of 4, but a 

 1-2 inch would require 16; the perfection of the image in the 

 latter case being very much less than in the former. 



It would be useless to attempt to assign by way of example 

 numerical values of the constant u for different kinds of objectives, 

 and of the limit of U which would be consistent with a sufficient 

 perfection of the image, in order to compute theoretically the 

 amount of super-amplification which every objective would bear. 

 The circumstances on which the first two elements depend, are 

 much too complicated for a theoretical estimation of their influence 

 in regard to the actual performance of the Microscope. Neverthe- 

 less, the foregoing considerations indicate the aim of the problem, 

 which is to determine the adequate focal length for every aperture. 

 It will be quite sufficient for our purpose to determine the limits of 

 admissible super-amplification directly by practical observations, 

 without further caring for the elementary conditions on which it 

 depends ; and if this is done, we have obtained all necessary data 

 for the problem under consideration. 



