DOUBLE REFRACTION OF CRYSTALS. 109 



generally conformable to this latter principle ; and for this rea- 

 son these hypotheses, though erroneous, represent the fact. 



If we put ^=the semiaxis of revolution of the ellipsoid of 



Huygens, o= its seraitransverse axis, o= the velocity of a 



ray of light in the interior of the crystal, and V= the angle its 



direction makes with the axis, the radius of the elipsoid will be 



a h 



V'a'^— (^a^— ^^). sin'\ V. Thus the velocity v, from the 

 principle of least action, being equal to unity divided by this ra- 

 dius, we shall havei'^=^— (^— ^). sin ^ V. 



This velocity is least when the ray of light is perpendicular 

 to the axis of the crystal, and then it becomes ^ : it is least, 

 when it is parallel to this axis, and then it is equal to ^. 

 Huygens found by experiment, that h is the ratio of the sine Connexion be- 



of refraction to the sine of incidence in the common refraction 'ween the two 



retractions, 

 of the Iceland crystal. This very remarkable result, which 



connects the ordinary and extraordinary refractions, is 'a neces- 

 sary consequence of the modifications that distinguish the ordi- 

 nary from the extraordinary ray not being absolute, but solely 

 relative to the position of the ray with respect to the axis of the 

 crystal. To show this, let us refer to the singular phenome- 

 non, that light exhibits after its passage through a crystal. 



In passing through a crystal the light is divided into two pen- Phenomena of 

 ., ,. , , ,. , I r I %ht after it* 



cils, one ordmary, the other extraordinary, and each ot them passage 



issues out of the crystal undivided, If we conceive a second through a 

 crystal placed beneath the first, in a situation perfectly similar, ^^^ * " 

 the ordinary ray will be refracted ordinarily on passing into the 

 second crystal, and the extraordinary ray will be refracted extra- 

 ordinarily. This will take place generally, if the principal 

 sections of the two opposite faces be parallel. By the principal 

 ^eofiow of a face is meant a section of the crystal by a plane 

 perpendicular to that face, and passing through the axis of the 

 crystal. But, if the principal sections be perpendicular to each 

 other, the ordinary ray will be refracted extraordinarily on pas- 

 sing into the second crystal, and the extraordinary ray will be 

 refracted ordinarily. In the intermediate positions, each ray 

 will be divided into two others at its entrance into the second 

 crystal. 



Now suppose a ray refracted ordinarily by one crystal to fall 



per- 



