DETERMINATION OF THE PATH OF KAYS. 19 



For the completion of the analogies which exist between the optical 

 action of a system and the case of refraction at one surface, the equi- 

 valent significance of the quantity /,; for the former and U Q or u for 

 the latter must here be further emphasized. It may be shown that 



n n* 



the relation expressed in (20), that is k = -- - = -_ # , obtains 



J J 



in a manner precisely analogous for u and u' t and that if we 

 represent the focal lengths of the first refracting surface by 

 cf> and <f>' and those of the second by $" and </>'", then 



u 



For if, in the equation for the direction of the ray after the first 

 refraction, we make (x N) = </>', and, consequently, x equal to 

 the abscissa of the focal point, and if we substitute for fi' its 

 value /3 + u%, then 



and, since this value must be for rays incident parallel to the 

 axis (for which /3 = 0), there results 



consequently 



Similarly, we get from the equation for the incident ray, if for 

 we put its value j3' ^&, 



It will be clear, without further explanation, that the ratios of 

 u' to $" and $" may be deduced in a similar manner. 



r ll 



The quantities w, u', k are, consequently, all equal to -- - , if / 



is the anterior or the posterior focal length, and n the index of 

 refraction of the medium ' in which the ray moves, before the 



c 2 



