On the Estimation of Aperture. By Prof. E. Able. 23 



ocular. When this diameter is greater than, or at least equal to, the 

 diameter of the eye's pupil, the brightness of the image has its 

 maximal value, which can never be increased, and is the same 

 brightness which would be obtained with direct vision by the naked 

 eye, of any large object under the same illumination ; and when 

 the ultimate diameter of the emergent pencil is the Z;th part of the 

 pupil's diameter, the brightness of the image is the ^'■^th part of the 

 brightness of unaided vision. 



Denoting by A the conventional distance of distinct vision, by 

 N the linear amplification of the image projected to this distance, 

 A 

 ^ = (^ will be the equivalent focal length of the total Microscope. 



If then a is the numerical aperture of the admitted pencil (which 

 may utihze either the whole aperture or a part of it only), the 

 diameter h of the ultimate emergent pencil at the plane of the 

 Eamsden circle will be according to proposition (7) of Sec. I. 



8 = 2 a <^, 



which is the diameter to be compared with that of the pupil in 

 order to obtain (by the squares) the ratio of the brightness of the 

 microscopical image to the brightness in vision with the naked eye. 



The difierent pJwtometrical equivalent of equal angles in dif- 

 ferent media, may be plainly demonstrated by several observations 

 which are already well known, and within the reach of every micro- 

 scopist, but I may briefly indicate some of them here. 



(1) Objects are seen with equal brightness, with the naked eye 

 and with the Microscope, whether they are uncovered or protected 

 by a covering-glass cemented on, provided their pellucidity is not 

 changed by the surrounding medium. (No such change takes 

 place, for instance, with perfectly transparent portions or elements 

 of a preparation.) It is evident that the pupil of the eye, or the 

 objective of the Microscope, admits from every radiant in air a 

 wider angular pencil than from the radiant in balsam, as the latter 

 pencil acquires the angular width of the former by an expansive 

 refraction at the surface of exit. The diameter of the object under 

 the covering-glass is not of course reduced by this refraction, but 

 appears of the same size still, and consequently the narrower 

 pencils emanating from the object in balsam must convey the same 

 quantity of light as the broader pencils emanating in air. 



(2) When a hemisphere of glass is cemented to a preparation 

 and the condition above referred to is fulfilled, the object appears 

 just as bright as it appeared uncovered, as well with the Microscope 

 as with the naked eye. In this case the divergence of the pencils at 

 their exit into air is not changed, and the pupil of the eye or the 

 lens-opening receives equal pencils under both chcumstances. 

 But as the hemisphere amplifies the object at its centre in the 



