DOUBLE REFRACTION. 



71 



diameter of the circle. In this case the plane of incidence 

 cuts the ellipsoid in a diame- 

 tral plane ; the points of con- 

 tact, and E, are both in that 

 plane ; and the extraordinary 

 ray observes the first law of 

 Descartes. Again, by the known 

 property of the ellipse, the points 

 of contact, E and 0, are on the 

 same right line, POE, perpendicular to the axis ; and are 

 to one another in the constant ratio of the axes of the ellipse. 

 Hence 



But 



OP 

 EP 



_ 

 EP ~ a 



tan OOP 



tanr 7 



tanECP tanr' 



We have therefore the remarkable relation between the 

 angles of refraction of the two rays 



tan r f b n' 

 tan r a n' 



This relation has been verified experimentally by Malus ; 

 and it is thus proved that the section of the surface of the 

 extraordinary wave by any diametral plane is an ellipse- 



The relation between the angles of incidence and refrac- 

 tion of the extraordinary ray in this case is easily deduced. 

 For 



tan r = 



sin r 



sin i 



V 1 - sin 9 r V ri* - sin 2 i 



Wherefore 



n' . . 



. sm i 



, n f n 



tan r' = tan r = . 



n V n* - sin 2 i 



