Some Bemarks on the Apertometer. By Prof. E. Ahhe. 25 



It withdraws the basis even of an indisputable indication of aperture, 

 making the angle to depend on two practically inaccessible elements 

 of very uncertain character — the refractive index and the external 

 curvature of the front lens. Crown glass is a variable thing. In 

 the crown fronts, for instance, of Mr. Zeiss' different objectives, the 

 refractive index of the ray D varies from 1 "501 to 1 "544, i. e. in 

 the range of 3 per cent, of the average value, and nobody, except 

 the maker, can know what may be the index in any given objective 

 without dismounting it. 



But there are many low-power object-glasses the front lens of 

 which is made of flint. Ought an interior angle of, say, 37° in such 

 a flint front of 1 • 62 refractive index to be considered as denoting 

 less aperture and less resolving power than 40° "interior angle" in 

 a crown front of 1 * 50 ? Moreover, what would be the signiflcation 

 of " first interior angle " if the front lens has not a plane external 

 surface, but a curved one ? I have at my disposal a dry J- objective, 

 made,for a special purpose, the numerical aperture of which is "91 

 (air angle = 132°) ; but the first interior angle, within the crown 

 front of 1"513 refractive index, is 87°, owing to a concave surface 

 of the front lens. If this objective were to be compared with other 

 glasses of ordinary construction by the interior angle, and if this 

 angle, in fact, had the prominent signification which Dr. Woodward 

 proposes, the glass spoken of would range among immersion lenses, 

 which, of course, neither Dr. Woodward nor anybody eke will 

 concede. 



I may briefly signalize the principal conclusions by which, in my 

 opinion, the product w . sin iv = a (a, numerical aperture ; tv, semi- 

 angle of aperture; 7i, refractive index of the external medium to 

 which the angle relates) can be demonstrated as the true and 

 general measure of aperture. 



1. The expression of aperture, as a quanfitij, must be based on 

 the evaluation of the number, or quantity, of rays collected by the 

 objective from one point of the object and transmitted to one point 

 of the image. 



No ray can exist in the wide-angled cone of light, emanating 

 from the object, which is not contained in the contracted cone of 

 hght going from the objective to the microscopic image, et vice 

 versa. As soon as this latter cone of light is reduced to a narrow- 

 angled pencil, owing to the distance of the apex, it is unquestionable, 

 from well-established optical })rinciples, that double or trijdo tho 

 angle represents double or triple the number of rays contained in 

 one section plane through the axis. Therefore the capacity of an 

 objective of collecting rays, i. e. its aperture, is proportionate to 

 tho angle of tho narrow-angled pencil on the side of the image, 

 other circum.stanccs being equal. 



On a theorem announced long ago by Professor llclmholtz and 



