SUPPLEMENT, 



&c. &c. 



FUNDAMENTAL FORMULA OF OPTICAL SYSTEMS, OR EQUATION OF THE 

 CHARACTERISTIC FUNCTION. 



1. The fundamental formula that we shall employ in our inves- 

 tigations respecting the geometrical properties of optical systems of 

 rays, straight or curved, ordinary or extraordinary, which, after issu- 

 ing from any luminous origin, have been any number of times 

 reflected and refracted by any combination of media, according to 

 any laws compatible with the known condition of least action, is the 

 following : 



3yW. = j-5x+ j^Sy + j-3.. (A) 



In this equation, w, y, z, are the coordinates of any point of the sys- 

 tem, referred to three rectangular axes ; a, ^, 7, are the cosines of 

 the angles which the tangent to the ray at that point, or the direc- 

 tion of the element ds, makes with the axes of coordinates ; v is the 

 quantity which in the hypothesis of molecular emission represents 

 the velocity of this element, and is supposed to be in general a func- 

 tion of the six quantities, .r, y, «, «, /3, y, depending on the nature of 

 the medium, and involving also the colour of the light; the partial 

 differential coefficients, 



S» 3y Si> 

 he' Ifi' 1;' 



