Crystalline Reflexion and Refraction. 161 



the direction of the vibrations is always perpendicular to that 

 assigned by Fresnel. And since, in order to make his views 

 agree with the phenomena, Fresnel was obliged to say that, in 

 an ordinary medium, the vibrations of a ray polarized in a cer- 

 tain plane are perpendicular to that plane, it is clear that, on the 

 present principles, we must come to a different conclusion, and 

 say that the vibrations of a polarized ray are parallel to its plane 

 of polarization. 



Conceive an ellipsoid with its centre at 0, the common origin 

 of the co-ordinates a?, y, z, #', /, z' ; and let its semiaxes be pa- 

 rallel to a, y, z, their lengths being equal M 



to -, T , - respectively. From the iden- 

 a b c 



tity of the condition (7) with that 

 marked (G') in Lemma III., it is evi- 

 dent that the directions of x f and y f , when 

 they are the two directions of vibration, 

 coincide with the axes of the ellipse in 

 which the plane of x'y' intersects the el- 

 lipsoid ; and if the right line OR, meet- 



Q 



Fig. 20. 



ing the ellipsoid in R, be the direction of x, we have 

 1 



or, by (8), 



(OR)'' 



a? cos 2 a + b z cos 2 /3 + c z eos 2 y, 



1 

 OR~ S ' 



so that OR is the reciprocal of the velocity with which the vi- 

 brations parallel to y' are propagated. Thus we see that the 

 vibrations parallel to either semiaxis of the elliptic section are 

 propagated with a velocity which is measured by the reciprocal 

 of the other semiaxis. 



Again, conceive an ellipsoid with its centre at 0, and its 

 semiaxes parallel to #, y, z, as before, but equal to a, b, c re- 

 spectively. Let this ellipsoid be touched in the point Q by a 

 plane which cuts OR perpendicularly in P, and draw the right 

 lines OP, PQ. Then as the condition (7) is identical with that 



ivr 



