FOR A NARROW BEAM OF LIGHT. 



383 



For the actual course of the ray the value of V , R , must be stationary as 

 regards variations of f,, ij lt , 7j t . Hence, differentiating with respect to these 

 variables, we obtain the four equations 



if. 



fli + 7 



*! 



! 

 = /*!- 



X, 



Since OPQR is a ray of the system, the co-ordinates x lt y,; f,, ^; |,, ^ 4 ; 

 x t , y., vanish together. Hence f 1 = g 1 =f t = g t = Q, and if we write A for the deter- 

 minant 



(8). 



Then 



(9). 



Substitutuig in (6), we obtain 



^o'fi- = VOE + W* + W> + iaA 1 + Si 



I 



