SECOND SUPPLEMENT. 



Statement and Integration of the Partial Differential Equation^ which 

 determines the Characteristic Function of Ordinary Systems of Rays, 

 produced by any Number of successive Reflexions or Refractions. 



1. Suppose that rays of a given colour diverge from a given 

 luminous origin, and undergo any number of successive changes of 

 direction, according to the known laws of ordinary reflexion and 

 refraction, at surfaces having any given shapes and positions, and 

 enclosing media of any given refractive indices. Let a, /S, y, be the 

 cosines of the angles which the direction of a final ray makes with 

 three rectangular axes, and let a;, y, z, be the three rectangular 

 coordinates, referred to the same axes, of a point upon this final ray ; 

 then u, jS, y, will in general be functions of a;, y, z, such, that if «. 

 denote the refractive index of the final medium, for rays of the given 

 colour, the expression 



is equal to the differential of a certain function V, of which I have 

 shewn the existence and the meaning in former memoirs, and which 

 I have called the characteristic function of the final system. The de- 



