116 Mb GREEN's SUPPLEMENT TO A MEMOIR ON 



(29) dJ L _d^_ (»* - If gfr I (when a? = 0). 



Or neglecting the term which is insensible except for highly refracting 

 substances, 



( 3 °) tf^ _ A^ ( wh en * = 0), 



* where m = — is the index of refraction. 



These equations belong to light polarized in a plane perpendicular 

 to that of incidence, and as <p and <p t are insensible at sensible distances 

 from the surface of junction of the two media, we have, except in the 

 immediate vicinity of this surface, 



_ ety 

 du 



V = — -TV" . 



dx 



* 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 



yp- / = a t sin (a t x + by + ct), 



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



yf/ t = Sa ( sin n (a t x + 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 



a t x + by + ct. 



Or the same equations might have been immediately obtained from (17), without in- 

 troducing 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. 



