On the Estimation of Aperture. By Prof. E. Abhe. 389 



distance which is great in comparison with its diameter — as is 

 always the case in the Microscope — we have at the bach of the lens 

 the same circumstances as are m front when a telescope-ohjective 

 is considered. Consequently, the larger or smaller number of 

 emergent rays will be properly measured by the clear diameter ; and 

 as no rays can emerge which have not been taken in, this estima- 

 tion must apply at the same time to the admitted rays, — other 

 circumstances, in particular the distance of the radiant from the 

 lens, being equal. 



The question, however, will now arise, how is the difference of 

 these other circumstances on the microscope-lens to bs taken into 

 account ? 



A simple consideration shows at once that this is properly done 

 by taking the absolute diameter of the lens (or it? " opening ") in 

 proportion to tlie focal length. When two lenses have equal 

 openings but different focal lengths, they transmit the same 

 number of rays to equal areas of an image at a definite distance, 

 because they would admit the same number if an object were sub- 

 stituted for the image, that is if the lens were used as a telescope- 

 objective. But as the focal lengths are different, the amplification 

 of the images is different also, and equal areas of these images 

 correspond to different areas of the object, /rowi which the rays are 

 collected. Therefore the higher-power lens with the same opening 

 as the lower power, will admit a greater number of rays in all from 

 one and the same object, because it admits the same number as the 

 latter from a smaller portion of the object. Thus if the focal 

 lengths of two lenses are as 2:1, and one of them amplifies an 

 object N diameters, the other of shorter focal length will amplify 

 the object 2 N diameters with the same distance of the image. 

 Consequently the rays, which in both cases are collected to a given 

 field, say of 1 mm. diameter, of the image, are admitted from a 



field of c^ mm. in the first case, and of ^tvt ^im. in the second. 



N 2N 



If now the idea of aperture referred to the photometrical quantity 

 of light, the capacities of equal openings with different focal lengths 

 would of course be in the inverse ratio of the areas from which 

 equal quantities are admitted, and would then be in the direct 

 ratio of the squares of amplification. Inasmuch, however, as the 

 opening is estimated by the diameter and not by the area, the con- 

 sideration is confined to the rays which travel within one meridional 

 plane of the lens, and the same principle must be applied to the 

 fields from which the rays are admitted ; which must also be 

 estimated by their diameters. The higher-power lens in the ex- 

 ample given above therefore admits tivice as many rays as the lower 

 power, because it admits the same number from a field of half the 

 diameter ; and, in general, the admission of rays with difl'erent focal 



