INTENSITY OF REFLECTED AND REFRACTED LIGHT. 407 



9. We adopt the following notation in addition to that already used : 



a, /3 are the motions parallel to x and ?/ of a particle in the upper medium. 

 a, 13, do. do. in the lower. 



I, R, T are the incident reflected and refracted vibrations. 

 I^ and T/ the corresponding normal motions. 



Occasionally Sai and 8?/ will be replaced by 



8 of cos <^ + 8'if sin 

 ^y cos <p — 8x'siD.<p 



respectively, when combined with a function depending on the incident wave, 

 and by 



8 a/' cos (p + 8^' siacp 



8 x" sm<p~-8i/' cos cp 



when combined with one which depends on the reflected wave. 



From the values of ^^ and 8p, it is clear that the axis of x' is the line of 

 transmission at incidence, and that of a/' at reflexion. The values of I, R, are in 

 general not required, but for the purpose of fixing the ideas, they may be con- 

 ceived to be as follows : 



1=0! cos (e X -vfy + ct) 



R = bcos( — ex+/y + ct+g) 

 T = c cos (e^ X +fy ■\-ct + K) 



\ =Ae-^^cos{fy + ct + n) 



T = Ce-^''' codify + ct + h + n) 

 If it should be thought that these values belong only to a particular case, I 

 would remark that, from the linearity of our equations, the results which we de- 

 duce for one circular function, are equally true, mutatis mutandis, of a series of 

 such functions. 



10. The values of «, /3, «„ /3,, deduced from the figure, are : 



a = I — R sin ^ + I^ 

 i8 =(I + R)cos0 

 «, = Tsin0' + T, 

 iS, = T cos 0' 

 The equations of motion in the upper medium are : 



^ = 2 {(Pr + ^(8x8a+8y8^)}8JT8a 

 = 2 (^(pr + ^8xA 8a+2^8x8y8^ 

 g = ,(cpr^^8f)8^^2i^8x8y8. 



