DOUBLE REFRACTION OF CRYSTALS. ] 0/ 



these means the very remarkable connexion, that Huygens 

 found by experiment, between the ordinary and extraor- 

 dinary refractions of the crystal, is deiiionstrated a priori, 

 as a necessary result of the law of extraordinary refraction. 

 The velocity of the ordinary ray in the crystal therefore is Difference of 



alwavs greater than tha^ of the extraordinary ray, ihe difference ^^.^ velocities 



- o . . <^f "^s two 



of the squares of tlse t\\ o velocities being proportional to the rays. 



square of the sine of the angle that the axis forms with the 

 latter ray. ■ According to Huygens, the velocity of the extra- 

 ordinary ray in the crystal is expressed by the radius of the 

 ellipsoid itself; his hypothesis therefore is conformable to the 

 principle of least action : but it is remarkable, that it is also con- 

 formable to the principle of Fermat, which consists in this. Principle of 

 that the light arrives from a given point without the crystal to Fermat. 

 a point within in the shortest tiixie possible ; for it is fa-y to 

 see, that this principle is reduced to that of the least action, by 

 reversing the expression of the velocity. Thus the-^law of 

 refraction given by Huygens is deducible equally from both of 

 these principles. For the rest, this identity of the laws of 

 refraction, deduced from the mode in which Huygens viewed 

 the refraction of light, with those given by the principle of 

 least action, takes place generally, whatever be the spheioid, 

 the radii of which, according to him, express the velocity of the 

 light in the interior of the crystal. This I demonstrate very 

 simply in the following manner. 



Huygens considers a ray R C, pi. II, fig. l6, falling on the 

 natural or artificial face A FE K of an Iceland crystal. Draw- of Huygens. 

 ing a plane, C O, perpendicular to this ray, and taking OK, 

 parallel to C R, to represent the velocity of light in vacuo, he 

 supposes, that all the points Co 6 O of the luminous wave ar- 

 rive in the same time, and in parallel directions, at the plane 

 K i' i I } which he finds thus. A F E D is an ellipsoid of revo- 

 lution, of which C is the centre, C D the semiaxis of revolu- 

 tion ; and the radii of which represent, according to Huygens, 

 the respective velocities of the light that follows their direc- 

 tions. Through the ray R C he draws a plane perpendicular 

 te the face, and cutting it in the right line B C K j and through 

 the point K he draw.s, in the plane of the face, K Tj perpen- 

 dicular to K C. Lastly, through K T he draws a plane K I, 

 touching the ellipsoid in I. Aceording to him C I is the direc- 

 tion 



