On the Estimation of Aperture. By Prof. E. Ahhe. 3 



lengths (the opening being the same) must be in the inverse ratio 

 of the focal lengths. 



In a single-lens Microscope, aperture must be determined, there- 

 fore, by the ratio between the dear ojyemng and the focal length 

 of the lens, in order to define the same thing, as is denoted in the 

 telescope by the absolute opening. 



(2) Regarding now composite systems — the most important case 

 in the Microscope — the further question arises, what is the opening 

 of such a system ? The actual opening, which limits physically the 

 transmission of the light through a composite objective, varies 

 according to particular circumstances. It may be the margin- of 

 the front lens, or of any one of the posterior lenses, or it may be a 

 diaphragm inserted in some part of the system. As the cone of 

 admitted rays expands continuously from the radiant up to the back 

 lens, the same objective admits of innumerable different openings 

 of this kind, which nevertheless may indicate the same aperture, 

 and thus no definite opening could be assigned. This ambiguity 

 cannot be removed unless we adhere to the diameter of the 

 admitted cone at that plane where it has its ultimate maximal 

 value, which is obviously the diameter of the pencil at its emer- 

 gence from the system, or, practically, the clear effective diameter 

 of the bach lens. The emergent pencil from a microscope-objective, 

 converging to a relatively distant focus, has its rays approximately 

 parallel, and the conditions are once more similar to those of the 

 telescope-objective on the side of the object. The diameter of this 

 emergent pencil, whether it emerges from a single lens or from a 

 composite system, must therefore always have the same signifi- 

 cation. 



The influence of the power or focal length also remains the 

 same as in the case of the single lens. An objective with a focal 

 length equal to half that of another admits with the same linear 

 opening twice as many rays as the latter, because the amplification 

 of the image at one and the same distance is doubled, and the same 

 number of rays, consequently, are admitted by the higher power 

 from a field of half the diameter. This must hold good, ivhether the 

 medium at the object is the same in the case of both objectives, or 

 different. For an immersion system and a dry system always give 

 the same amplification when the focal length is the same. 



Thus we have as general propositions for all kinds of objectives : 

 (a) the admission of the rays with one and the same power or 

 focal length varies with the linear diameter of the pencil at its 

 emergence ; (b) with different powers, the same admission requires 

 different Hnear openings in the proportion of the focal lengths— 

 or conversely, the admission by one and the same opening is in 

 inverse proportion to the focal length. Consequently the aperture 

 of an. objective is always exhibited by the ratio between the linear 



B 2 



