which function the properties of the system depend. In the appUca- 

 tions of this formula, to systems of straight rays, ordinary or extra- 

 ordinary, it is advantageous to introduce the consideration of a 

 characteristic function of another kind, depending on the direction 

 rather than on the coordinates of the ray, but connected with the 

 former function, and with the geometrical properties of the system, 

 by relations which form the chief subject of the present Memoir. 

 The theory of these relations, from the generality of its nature, will, 

 perhaps, be interesting to Mathematicians: I am aware that it admits 

 of being much farther extended, and that much remains to be done, 

 in order to render it practically useful. 



WILLIAM R. HAMILTON. 



Observatory, ^pril, 1830. 



