On the Estimation of Aperture. By the late C. HocJcin, jun. 345 



ence of sensible heat from one body to the other, but the nature of 

 the radiation is not of importance to the reasoning, and what is 

 true of radiant heat is true of Mght in the case in question. 



It follows that an object under given illumination will radiate 

 more light when in a medium of refractive index n than in one in a 

 medium of lower refractive index n in the proportion of n^ : n!'^. 



This leads to the only point not considered in the preceding 

 investigation. 



The aperture was measured by the angular breadth of the 

 pencils forming the image multiplied by the number of pencils. 



The absolute intensity of the light in each pencil was not 

 considered. 



It is evident from equation A that if I is the intensity of 

 normal radiation of the object, the light in a small central incident 

 pencil of angular breadth a is measured by a number propor- 

 tional to I a^ and this pencil has on emergence the smaller 



angular breadth :rj: - a, and the intensity of the light is propor- 



/ ^' \2 

 tional to I ( N - « I. But as I itself has been shown to vary as n"^ 



the intensity becomes proportional to W. a? as n' = 1, or for a given 

 magnifying power it is constant. 



The result is that Prof. Abbe's formula squared does give the 

 true aperture of the instrument measured by the amount of light 

 used to form the image of a plane object under fixed illumination. 



The same formula is also proportional to the resolving power of 

 the lens, for consider a series of dark and transparent bands of 

 small equal breadth ^l, on sl glass plane illuminated from below. 



The light in any direction A P coming from A is reinforced by 

 that coming from B when A P — P B is equal to a whole number 

 of wave-lengths (fig . 5). This is first the case when A P — P B 

 is one wave-length, say \, or when A B sin w = A, = Z sin m. 



But A, varies inversely as the refractive index of the medium 



and is equal to \' - say. 



Then \' = lnaiau or I 



A.' 



therefore the greater n sin u may be, the less may I be, in order 

 that the first pair of difi'racted rays may enter the objective, and so 

 form with the central pencil a pencil of finite angular breadth 

 emergent from the objective and incident on the eye-piece, through 

 the image of the object giving a defined representation of the 

 object. 



