472 Lord Rayleigh's Notes on some 



where W now measures the source at 0', and ^r the luminous 

 amplitude at 0. But by the reciprocal theorem 



yjr' :"^' = i|r:^ 



and thus 



<t x a 2 ' 



or the " apparent distance " of 0' from is the same as of 

 from 0'. 



If we now assume the reciprocal character of apparent dis- 

 tance, there is no difficulty in deducing the law of apparent 

 brightness in a perfectly general manner. For consider the 

 whole light received over a small area cr x at 0' (perpendicular 

 to the ray) from a small luminous area a 2 at (also perpen- 

 dicular to the ray). If, as before, ay 1 denote the angular 

 opening at of the pencil which corresponds to a l9 this light 

 is proportional to coicr 2 . But (o 1 a- 2 = «2°"i? if °>2 ^ e the appa- 

 rent magnitude of c 2 seen from 0'. Hence the whole quantity 

 of light received by o^ is proportional to (s& 2 <J\ ; so that if c^ 

 be given, representing, for example, the area of the pupil, 

 the whole light received at 0' from a small area <r 2 at is 

 proportional to the apparent magnitude of that area. In 

 other words the apparent brightness is constant. 



In this way of regarding the matter the law of apparent 

 brightness becomes a deduction from the general reciprocal 

 theorem. The argument may, of course, be reversed, so as 

 to exhibit the reciprocal character of apparent distance as a 

 consequence of the law respecting brightness. And this view 

 of the subject may perhaps commend itself to those who 

 appreciate the independent evidence for the law of brightness 

 derived from the theory of enclosures as based upon the 

 second law of thermodynamics. 



In any case the law connecting magnifying power with ihe 

 section of the pencil follows as an immediate consequence. 

 If cr 2 be given, its apparent magnitude, as seen from 0', is 

 given by 



ft)! 

 ft) 2 — (T 2 — , 



<?1 



and is, therefore, inversely proportional to the section at 0' 

 of a pencil of given angular magnitude issuing from 0. This 

 principle is of great use in the design of optical instruments. 

 The application to the telescope is fully stated by Smith. By 

 means of it the one- dimensional magnifying power of prisms, 



