IN PERFECTLY TRANSPARENT MEDIA. 205 



equations is very simple. We have necessarily either /> 2 = 

 V^Uj 2 , or ^ = and V 2 =U 2 2 . In this case, the light is linearly 

 polarized, and the directions of oscillation and the velocities of 

 propagation are given by Fresnel's law. Experiment has shown that 

 this is the usual case. We wish, however, to investigate the case in 

 which does not vanish. Since the term containing arises from 

 the consideration of those quantities which it was allowable to neglect 

 in the first approximation, we may assume that is always very 

 small in comparison with V 3 , Uj 3 , or U 2 3 . 

 17. Equations (28) may be written 



V2_U 2 =-t^^ V 2 -U 2 --b-^ &. (29) 



u i " t V/o 1 ' 2 " V P 2 



By multiplication we obtain 



V2(V 2 - U^X V 2 - U 2 2 ) = 2 . (30) 



Since is a very small quantity, it is evident from inspection of this 

 equation that it will admit three values of V 2 , of which one will be a 

 very little greater than the greater of the two quantities t^ 2 and U 2 2 , 

 another will be a very little less than the less of the same two quan- 

 tities, and the third will be a very small quantity. It is evident that 

 the values of V 2 with which we have to do are those which differ but 

 little from U* and U 2 2 .* 



For the numerical computation of V, when U x , U 2 , and are 

 known numerically, we may divide the equation by V 2 , and then 

 solve it as if the second member were known. This will give 



(31) 



By substituting T^Ug for V 2 in the second member, we may obtain 

 a close approximation to the two values of V 2 . Each of the values 

 obtained may be improved by substitution of that value for V 2 in the 

 second member of the equation. 



For either value of V 2 , we may easily find the ratio of p l to /o 2 , 

 that is, the ratio of the axes of the displacement-ellipse, from one of 

 equations (29), or from the equation 



n 2 V2 _ TT 2 



2__i_ u i (32) 



n 2-V2_TJ 2 

 Pl U 2 



obtained by combining the two. 



* We should not attribute any physical significance to the third value of V 2 . For 

 this value would imply a wave-length very small in comparison with the length of 

 ordinary waves of light, and with respect to which our fundamental assumption that 

 the wave-length is very great in comparison with the distances of contiguous molecules 

 would be entirely false. Our analysis, therefore, furnishes no reason for supposing that 

 any such velocities are possible for the propagation of electrical disturbances. 



