DETERMINATION OF THE PATH OF RAYS. 13 



or, if we substitute for /3' and b* their values from (6) and (4), 



. 



If we ta'ke, for the sake of brevity, 



n 

 then the above expression becomes 



or 



Similarly, we obtain from equation (4), if for /3 X and N* 

 we substitute their values, 



To further simplify these expressions let 



% r + i = i 



consequently 



(10) 



Conversely, the quantities /3 and & can, of course, be determined 

 through /3* and &*, if the latter are known. It is only necessary, 

 for this purpose, to substitute successively the values of b and /3, 

 obtained from the one of the two equations, in the other equation, 

 and in the reduction to bear in mind that g I h k 1. This 

 latter ratio may be readily proved by multiplication of the cor- 

 responding expressions in (9). We obtain 



= /ft* _ 7,/9* ( ' 



If P is a given point on the straight line (if necessary produced) 



