10 



function V, the characteristic function of the system; and the funda- 

 mental formula {A), that expresses its variation, namely : 



we shall call the equation of the characteristic function. 



Other Characteristic Function for Systems of Straight Rays. 



4. In the remaining reasonings of the present Supplement, we 

 shall confine ourselves to the consideration of homogeneous systems 

 of straight rays not parallel ; and in investigating the properties of 

 such systems, it will be useful to employ another function, connected 

 with the function V by many remarkable relations. This new func- 

 tion, which we shall call W, is determined by the condition : 



„^ , ,, Ju . 3i) . 3u „ 



which gives, on account of {A), or (C), 



3^=.3^ + y3|+.3^ (E) 



It results from this differential equation (E) (in which we employ 

 the sign of variation d to mark the connexion with the definite 

 integral yvds, a remark which applies to the whole of the present 

 Supplement,) that if the variations of x^ y, «, be such as to leave 



a, /3, y, and consequently 



Sx) 3u 3t) 

 3» ' 3/3 ' Jy 



unchanged, that is, if we pass from any one point of the system to 

 any other point situated upon the same ray, the function W will not 

 vary. We may, therefore, consider W as a function of «, /3, y, of 

 which the form can be determined from that of F, by eliminating 

 .T, t/, z, between the equations (C) and (D), when the nature of the 



