286 ON THE KEFLEXION AND REFRACTION OF LIGHT. 



___ _ . .(30), 



dx dx 



* where JJL = is the index of refraction. 



These equations belong to light polarized in a plane perpen- 

 dicular to that of incidence, and as <f> and <, are insensible 

 at sensible distances from the surface of junction of the two 

 media, we have, except in the immediate vicinity of this sur- 

 face, 



dty 

 u == i 



y I (31). 



" dx 



When light is polarized in the plane of incidence, the con- 

 ditions at the surface of junction have been shewn to be 



10=10, i 



dw dw t f- (when a? = 0) (32). 



dx dx J 



Since in these conditions we may differentiate or integrate 

 relative to any of the independent variables except a?, we see 

 that the expressions (30) and (32) are reduced to a form equiva- 



* Though these equations have been obtained on the supposition that the 

 vibrations of the media follow the law of the cycloidal pendulum, yet as (b) has 

 disappeared, they are equally applicable for all plane waves whatever. 



In fact, instead of using the value 



\ff t = a, sin (a/B + by + ct), 

 and corresponding values of the other quantities, we might have taken the infinite 



series 



\p, = Sa y sin n (ax + by+ ct), 



where a and n may have any series of values at will. But the last expression is 

 the equivalent of an arbitrary function of 



0,05 + by + ct. 



Or the same equations might have been immediately obtained from (17), 

 without introducing this consideration. The method in the text has been employed 

 for the sake of the intermediate result (29), of which we shall afterwards make use. 



