The Relation of Aperture and Power. By Prof. Abbe. 461 



i. Relation of Aperture to Power in regard to the entire 

 Microscope. 



. 1. The first question may be dealt with under these two 

 heads : — (a) What are the smallest dimensions of microscopical 

 detail which are within the reach of any given aperture ? (b) What 

 visual angle is required for the distinct recognition of details of 

 given dimensions ? If these can be answered in a reliable manner, 

 the first question will have been disposed of. 



The smallest dimensions which are within the reach of a given 

 aperture are indicated with sufficient accuracy by taking the limit 

 of the resolving or separating power of that aperture for periodic 

 or regular structures, i.e. the minimum distance apart at which 

 given elements can be delineated separately with the aperture in 

 question. The numerical expression of that minimum distance is 



» = ji, 



a 



where a denotes the numerical aperture and A, the wave-length of 

 light ; a fair average is obtained for the latter element (with 

 observations with the eye and white light), by taking \ = ' 55 fi 

 = - 00055 mm.; i. e. the wave-length of green rays between 

 the lines D and E, very near to the point of maximum visual 

 intensity in the diffraction spectrum. 



Though this expression applies in strictness only to the visi- 

 bility of periodic structures composed of regularly arranged 

 elements, it may be taken as an approximate measure of delinea- 

 ting power in general, i. e. in regard to structures of every 

 composition. My theoretical investigations and experiments show 

 that with objects of every shape and arrangement, the micro- 

 scopical image will not present any indications of structure, the 

 dimensions of which are perceptibly below the value of 8, given 

 (for any aperture) by the above formula. Prominences of any 

 shape on the outline of a coarser object, for example, will disappear 

 more and more as their dimensions approach the value of 8 for the 

 aperture in use. Isolated elements of triangular, Gr quadratic, or 

 rectangular figure will look more and more alike (becoming more 

 and more circular or elliptical in form), as they diminish in size 

 to the value of 8* The loss of diffracted light attendant upon 

 the limitation of the pencils by the lens-opening, changes or 

 obliterates those details which are beyond the limit of the resolving 

 power. Consequently the microscopical image of an object of 

 any composition whatever, will always be dissimilar to the object 

 to the same extent ; and the limit of resolving power therefore 

 indicates the limit of similar or correct delineation generally. 



* I have verified these theoretical inferences, for small apertures, by many 

 experiments, with objects of very different nature. 



