On the Estimation of A;perture. Bij Prof. E. Able. 9 



(4) Equal angles of the admitted pencils from radiants in 

 different media do not yield equal apertures, but apertures which 

 are in the exact ratio of the refractive indices of those media. 

 Thus the diameter of the emergent pencil of an immersion glass 

 which takes in a cone of say 120° from an object in balsam, is 

 greater in the proportion of 3 : 2 than the diameter of the emergent 

 pencil of a dry lens of equal power admitting the same angle from 

 an object in air. Attentive microscopists and opticians have long 

 since noticed the fact, that immersion objectives require and utilize 

 much larger back lenses than equal-power dry systems of similar 

 aperture-aw^Ze. 



(5) An immersion objective may have a greater aperture than 

 any dry lens of even 180° aperture-aw^Ze can have. The maximal 

 opening of a dry lens (i. e. the maximal diameter of the pencil 

 emergent from such a lens) is shown by proposition (7) to be 

 exactly double its focal length, for as p = f (n sin ti) and w = 1 

 and sin w = 1 for air, p = / or (for the whole diameter) 2 p = 2/. 

 No lens performing on objects in air (n = 1) can therefore ever 

 admit of a wider aperture, because no angle u is possible whose sine 

 is >1. When, however, the object is in a denser medium (and 

 no film of air with plane surfaces is between that medium and the 

 system) an angle of aperture which is much less than 180° 

 (exceeding only the double of the critical angle for the medium) 

 will utilize and require a wider opening of the system than 2/. 

 The excess of the numerical aperture of an immersion glass beyond 

 the unit gives a direct expression of the surplus of aperture over 

 the maximal aperture of a dry lens of an air-angle of 180^. 



(6) The unit of aperture is exhibited by an objective which 

 gathers -in the whole hemisphere of radiant light in air. The 

 value of a for any given objective shows the capacity of that 

 objective in comparison with the capacity of another of maximal 

 air-angle. 



Any one who has not comprehended the generahty of the 

 demonstration, may object that the greater or less opening required 

 for the transmission of a pencil of given angle depends on the 

 particular mode in which that pencil is refracted by the lens- 

 surfaces of the system. A pencil of 120° in air requires, it will 

 be said, a smaller opening than the same pencil in balsam when 

 a homogeneous-immersion objective is used, because in a dry 

 lens it is contracted on its entrance into the system by the refrac- 

 tion of the plane front-surface, whilst the pencil in balsam, owing 

 to the abolition of the front-refraction by the immersion, is not 

 subjected to such contraction, and therefore maintains a greater 

 linear diameter up to the plane of emergence. 



The fallacy of such an objection is readily shown : Take a 

 front lens with a concave surface of admission, of such a curvature 



