On the Estimation of Aperture. By Prof. E. Abbe. 13 



meter. The proportion of tlie clear openings (or effective diameters) 

 with the object in balsam or in air may thus be strictly ascertained. 

 If the objective should be rated at, say, 1-20 num. ap., the ratio of 

 the diameters will always be found 6 : 5 (i. e. as 1 -20 to 1 -00) ; 

 and if, in another objective, the num. ap. should be 1-40, this ratio 

 will always be as 1 '40 to 1 • 00 or as 7 : 5.t 



The interpretation of this experiment is plain. In focussing an 

 immersion objective on an object with air above, it is obviously con- 

 verted into a true dry lens, the under surface of the covering -glass 

 acting as the plane front-surface of the system. If the covering- 

 glass is very close to the object the distance of the radiant from the 

 plane surface will be so small that an exceedingly small central 

 portion of this surface is sufficient for admitting to the front all 

 rays up to an obliquity of 88° to 89°. The objective then acts 

 as a dry lens of nearly 180° aperture-angle, and gathers-in almost 

 the whole hemisphere of light from the radiant in air ; whilst the 

 same systems when focussed on an object in balsam, admit no wider 

 cone (in the examples mentioned in the preceding paragraph) 

 than 108° or 138°, in fact much less in each case than a hemi- 

 sphere. Nevertheless, the emergent pencil of rays is much 

 narrower with the whole hemisphere of rays in air than it is with 

 the smaller cone of rays in balsam, whilst the amplification of the 

 image is not increased — the power of an optical system of any 

 kind whatever being exactly the same, whether there is refraction 

 or no refraction at its anterior plane surface. 



Every one will concede that there is a true reduction of aperture 

 when a brass-stop is inserted at the back of a given system, 

 stopping off a certain marginal zone of the clear opening which 



t According to the general proposition (7) the linear diameter of the reduced 

 opening of an immertiion glass with a dry object must be = 2/, provided the film of 

 air beneath the covering-glass be very thin. By measuring the reduced opening of 

 such an objective in the way suggested above, and taking its half, the exact focal 

 length of tlie system is obtained. 



The same principle may be made use of for objectives of every kind. When 

 the numerical aperture of an objective (or the numerical equivalent of any 

 smaller angle within the aperture-cone) is determined, and the linear diameter of 

 the corresponding emergent pencil at the plane of the posterior principal focus of tlie 

 system is measured micrometrically, the focal length is at once obtained from 

 formula (7). 



The author has for many years applied this very convenient and accurate 

 method for measuring focal lengths. 



On the other hand, the proposition (6) also indicates a new method for mea- 

 suring apertures and aperture-angles. When the amplification N of an objective 

 for a definite position of the image 0* is ascertained, by projecting the image of 

 a stage-micrometer upon an eye-piece micrometer, the auxiliary Slicroscope 

 may be focussed to any convenient plane and the linear diameter 2p of the 

 emergent pencil measured there. If now the distance / of that s>iine plane from 

 the image to which the amplification N relates, is measured likewise, we have all 

 the elements for computing the strict value of a = « sin m— and of the angle u 

 also— by means of formula (6). This method enables us to measure immersion- 

 apertures without requiring a disk of glass or similar devices. 



