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LV. On the Relation between the Brightness of an Object and 

 that of its Image. By W. T. A. Emtage, M.A* 



IF an object situated in a medium of refractive index /x. 

 forms an image by means of rays refracted through any 

 number of surfaces and finally into a medium of index yJ , the 

 luminosities, I, F, of object and image satisfy the relation 



supposing no light to be lost by reflexion at the surfaces or 

 by absorption in the media. 



This is generally proved, in the case where the pencils are 

 not all very narrow and along an axis of revolution of the 

 refracting surfaces, by showing that if this relation did not 

 obtain we could get an image brighter than the object by 

 viewing this through a suitable combination of lenses ; a 

 result which would be contrary to experience. The following 

 direct proof may, however, be given for pencils of any sort: — 



We notice first that if a small luminous surface sends a certain 

 quantity of light to a second small surface, for the second to 

 send the same quantity of light to the first it must be equally 

 bright with it. For if s 1} s 2 are the areas of the two surfaces, 

 I T , I 2 their luminosities, d the distance between them, and 

 0i, 2 the inclinations of their normals to d, the lights which 

 they send to each other are 



I] S] s 2 cos } cos 2 , I 2 s l s 2 cos X COS 0o 



~dT ~' and " ~oT~ ~ m 



And for these to be equal T 1 = T 2 . 



Now let s be a small area of an object in a medium of 

 refractive index fi ly and let it send out a sheaf of rays which 

 is refracted across the surface RT into a second medium of 

 index /* 2 . The cone of rays from s converging to the point 

 R is retracted into a cone of different solid angle. If i and r 

 are the angles of incidence and refraction at R, the angular 

 breadths of these solid angles in the plane of refraction are 

 di, dr ; and their angular breadths at right angles to this 

 plane are proportional to sin z, sin r. 



Now . . 



fi l sin i=fi 2 sin r, 



.". im l cos i di=fA 2 cos r dr. 

 * Communicated bv the Author. 



