DOUBLE REFRACTION. 69 



ratio of the sines of incidence and refraction, is constant. 

 Hence the ray CO observes both the laws of Descartes. On 

 the other hand, the point of contact with the ellipsoid, E, is 

 not, in general, in the plane of incidence, EOF ; and further, 



PT> 



the ratio is not constant. The extraordinary ray, there- 

 fore, does not in general follow either of the Cartesian laws. 

 There are certain cases, however, in which the extra- 

 ordinary ray observes one, or both, of these laws. The most 

 important of these is that in which the refracting surface 

 contains the axis of the crystal, and the plane of incidence is 

 perpendicular to the axis. In this case, the axis of the 

 crystal is a right line passing 

 through C, perpendicularly to 

 the plane of incidence ; and 

 the section of the ellipsoid by 

 that plane is its equator, and 

 is therefore a circle whose 

 radius is the greater axis of 

 the generating ellipse. The 

 tangent plane passing through F touches the ellipsoid in a 

 point of that circle ; and therefore the ray CE drawn to that 



CiT) 



point is in the plane of incidence. Again, ^, which is the 



ratio of the sines of incidence and refraction of the extra- 

 ordinary ray, is a constant quantity ; and therefore the 

 second law of Descartes is also observed. 



CE being the greatest value of the radius- vector of the 



P"R 



ellipsoid, the ratio -^ is, in this case, the least ratio of the 

 OJii 



sines of incidence and refraction for the extraordinary ray. 

 This least value is the extraordinary index. 



Let n and n f denote the indices of refraction of the ordinary 

 and extraordinary rays ; then 



