344 Transactions of the Society. 



illumination, it becomes evidently a rational expression for 

 aperture. 



Of course, in the comparison of lenses numerically, tlie square of 

 n sin u must be used just as the square of the diameter of an 

 object-glass is the true measure of its light-admitting power. 



It remains to be shown that the convention adopted for counting 

 rays is not merely one adopted because it yields consistent results 

 geometrically, but that it is founded on physical facts. For this 

 Prof. Abbe refers to Clausius. English readers will find Prof. 

 Clausius' memoir on radiation in the last memoir of his work on 

 Heat, edited by Hirst (J. Yan Voorst, 1867). The nature of 

 Prof. Clausius' argument may be illustrated thus. 



Eeferring to the diagram fig. 3, let P Q be a section of a small 

 circular disk radiating heat, p q the section of another disk similar 

 in all respects to the first and its " optical image." Suppose also 

 that both are at the same temperature. 



Let I be the intensity of normal radiation from P Q, and i be 

 the intensity of normal radiation from p q ; then the quantity of 

 heat sent from P Q to ^ ^ in unit time is measured by 



I (P Q . Z'' A P By, 



and that sent from p g- to P Q in the same unit time is 



i(pq Z.^Ap By. 



The ratio of these quantities is 



A VQ /' AvB / ' 



i\pq/_'ApB 



n' 



and this has been shown to be equal to - . -^ 



% n^ 



I n^ . 

 Now unless -r = — ^ this expression must difier from unity, and 



so a greater amount of heat would be sent from "P Q, to pq than is 

 sent from p q to V Q, or vice versa. 



In either case one of the bodies would be heated at the expense of 

 the other, and we should have in a short time a hot body heated by 

 a cooler one without the intervention of any mechanism doing work. 

 This would be contrary to the second law of thermodynamics, that 

 heat cannot of itself pass from a colder to a hotter body, a law 

 which is found to hold whenever tested by direct experiment, and 

 one which has never led to false conclusions when used in predicting 

 the phenomena that should result from given conditions, of what- 

 ever degree of complexity these may be. 



Prof. Clausius treats of radiant heat as the means of transfer- 



