98 



sign of the present Supplement, is to point out some new properties 

 and uses of this function, resulting from the partial differential 

 equation 



-which we obtain by eliminating the three cosines a, /3, y, between 

 the three equations 



dv dv . dv ,„> 



by the help of the known relation 



a' 



+ /3^ + 7= = 1 



2. The equation {A), is a particular case of a more general 

 differential equation, for all optical systems of rays ordinary or 

 extraordinary, obtained by eliminating the same three cosines, «,/3,7, 

 by the same known relation between the three following equations, 

 assigned in my former memoirs, 



dF_S«^ dV_lv_ rff_3u. 

 ^ "*~ ?« ' dy ~' i^' dz ~" Sy ' 



in which V is the characteristic function of the system, and t; is a 

 homogeneous function of «, /S, y, of the first dimension, representing 

 the velocity of the light, estimated on the hypothesis of emission, 

 and differentiated as if a, /3, y, were three independent variables. 

 And the integral of (^), is a particular case of a more general inte- 

 gral, extending to all optical systems of straight rays, and consist- 

 ing of the following combination of equations, assigned in my fbr- 

 mer Supplement : 



