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XXVI. On the General Laios of Brightness of Images. 

 By Prof. J. D. Everett, F.R.S., Queen's College, Belfast*. 



I AM not acquainted with any optical Treatise which con- 

 tains a general investigation of the brightness of an 

 image seen by rays which have traversed a medium of con- 

 tinuously varying index. The only discussion of the subject 

 that I have met with is contained in chapter xii. of Clausius's 

 ' Mechanical Theory of Heat.' That chapter is entitled " On 

 the Concentration of Kays of Light and Heat," and the main 

 purpose of it is not optical. I have found that its optical 

 results can be obtained by a much simpler method, as set 

 forth in the present paper. The results themselves are pro- 

 bably familiar to leading physicists, but they certainly are 

 not so well known as they deserve to be. 



The particular case in which all the refractions and reflex- 

 ions take place at coaxal spherical surfaces with nearly normal 

 incidence is treated in my edition of ' Deschanel ' (see espe- 

 cially footnote to § 768 of the earlier, or § 1037 of the later, 

 editions), and had been previously treated by a different 

 method in Helmholtz's ' Physiological Optics ' (pp. 171-175). 



The mode of investigation here adopted has been suggested 

 partly by Clausius's discussion above referred to, partly by an 

 investigation of Kirchhoff's in Poggendorff's Annalen, vol. cix. 

 pp. 284-287, to which Clausius refers, and partly by § 334 

 (2nd edition) of Thomson and Tait's ' Natural Philosophy.' 



We shall begin by establishing the following 



Theorem on Apparent Size. 

 If two small equal plane areas A 1} A 2 are so placed that a 

 ray from the centre of one to the centre of the other is perpen- 

 dicular to them both, the two solid angles formed by rays at the 

 centre of each which come from the circumference of the other are 

 inversely as the squares of the absolute indices of refraction at 

 the centres at which they are formed. That is, if fi x fi 2 be the 

 indices at the centres of A x A 2 , co t the angle formed at the 

 centre of A 1 by rays from the circumference of A 2 , and g> 2 

 the solid angle formed at the centre of A 2 by rays from the 

 circumference of A 1; then will 



fl l ^A l W l = fJL 2 2 A 2 C0 2 (1) 



As a particular case, if the two indices of refraction are equal, 

 as well as the two areas, the two solid angles will be equal ; 

 in other words, the apparent size of the first area as seen from 



* Communicated by the Author. 



