398 Transactions of the Society. 



IV. — Experimental Demonstration of the Aperture-equivalent. 



In the foregoing discussion I have demonstrated the aperture- 

 equivalent without regard to experiment and on general optical 

 principles only, as is necessary for an exhaustive scientific settlement 

 of the subject. A theoretical discussion of this kind is, however, by 

 no means required for demonstrating the essential principle of 

 " numerical " aperture. That there is an unequal equivalent of 

 equal angles in different media in regard to ajjerture, is a fact 

 which may be readily shown by observations of the most simple 

 character. I confine myself to a few examples which have been 

 referred to in previous discussions. 



(1) If any dry lens of an aperture-angle w for objects in air, is 

 focussed on a balsam-mounted object, with a plane surface of exit, 

 the aperture-aw^Ze at the radiant is of course reduced to a smaller 

 angle v, according to the condition 



. V . w 



nsm-= sin - , 



in which n denotes the refractive index of the balsam. It is, 

 however, clear that the amplification of the image is not changed 

 — the power of the system is the same still — whilst the linear 

 diameter of the emergent pencil remains the same also. Con- 

 sequently, the ratio of the opening to the focal length — i. e. the 

 aperture — is not reduced. 



This simple fact thus contains a direct proof of the proposition 

 that different angles in difierent media may denote equal aper- 

 tures. 



The idea of aperture being, as has been said, dependent on 

 that of " opening," the assertion that aperture is " cut down " by 

 the balsam, or by the immersion, is obviously an abuse of the 

 term, independently of the fact that the assertion is not supported 

 in any way by what we know as to the actual performance of 

 objectives with these " cut-down " apertures. 



(2) Moreover, suppose the same objective of w° air-angle to be 

 focussed on an object in balsam, the surface of exit, however, being 

 no longer a plane surface, but a spherical one, the object being at 

 the exact centre of a small hemisphere of glass or balsam ; — or 

 suppose the original objective to be provided with an extra immer- 

 sion-front, the centre of the curvature of which coincides with the 

 focus. In this case the angle of the admitted pencil will be the 

 same for the radiant in glass or balsam as it was for the radiant in 

 air ; and the clear opening will also be the same still. It would, 

 however, be obviously a mistake to say that the objective had now 

 undergone no change of aperture, or that the full aperture was 

 now made to bear upon a balsam-mounted object. For it is an 



