On the Estimation of Aperture. By Prof. E. Able. 5 



wide-angled system; u, U the angles of inclination of any two 

 rays admitted from the radiant, and u*, U* the angles of the same 

 rays on their emergence ; then we shall have always 



or 



sin U* : sin M* : : sin U : sin w ; 

 sin U* sin it* 



sin U 



sm u 



= const = c ; 



(1) 



i.e. the sines of the angles of conjugate rays on both sides of an 

 aplanatic system always yield one and the same quotient c, what- 



Fig. 1. 



ever rays may be considered, as long as the same system and the 

 same foci are in question. 



This proposition holds good for every composition of the system 

 (every arrangement of media and refracting surfaces), and for every 

 position of object and image. In point of fact, the laiv of con- 

 vergence for aplanatic systems, as indicated above, is the necessary 

 condition (physically and geometrically) on which depends the 

 delineation of an image by means of wide- angled pencils. When 

 in any case the convergence of the rays in a system of lenses is not 

 in accordance with this condition (very approximately at least) that 

 system will be unfit for depicting an image of an object, except by 

 warrow-angled pencils. 



Microscope-objectives do of course depict images with wide- 

 angled pencils, and consequently the proposition must apply to 

 them vnthout any restriction ; the author, it will be remembered, has 

 suggested a simple experiment f by which every one may satisfy 

 himself that all such objectives, if moderately well made, are in 

 perfect accordance with this statement. 



Suppose now, that for any particular objective and any parti- 

 cular position of the conjugate foci of object and image, the value 

 of the constant quotient c of formula (1) is determined numerically, 

 in any way whatever ; it will then be possible to compute the 



t See this Journal, iii. (1880) p. 511. 



