200 FRESNEL'S THEORY OF DOUBLE REFRACTION. 



a 3 , 6 2 , c 2 , being the elasticities in the directions of the three 

 axes ; r the radius- vector of the surface ; and a, ]3, 7, the 

 angles which it makes with the axes. 



This surface determines the velocity of propagation of 

 the wave, when the direction of its vibration is given. For, 

 the ethereal molecule vibrating in the direction of any radius 

 vector, r, of this surface, the elastic force which governs its 

 vibration will be proportional to r z ; and, since the velocity 

 of wave-propagation is as the square root of the elastic force, 

 it must in this case be represented by r. Hence, if we con- 

 ceive the vibration in the incident wave to be resolved into 

 two within the crystal, performed in two determinate direc- 

 tions, these will be propagated with different velocities ; and, 

 as a difference of velocity gives rise to a difference of refrac- 

 tion, it follows that the incident ray will be divided into two 

 within the crystal, which will in general pursue different 

 paths. Thus, the bifurcation of a ray, on entering a crystal, 

 presents no difficulty, provided we can explain in what man- 

 ner the vibration comes to be resolved. 



(212) To understand in what manner this takes place, 

 let us conceive a plane wave advancing within the crystal. By 

 the principle of transversal vibrations, the movements of the 

 ethereal molecules are all parallel to the wave. But the mo- 

 tion of each molecule, when thus removed from its position of 

 equilibrium, is resisted by the elastic force of the medium ; 

 and that force is, in general, oblique to the direction of the 

 displacement. If the plane containing the direction of the force 

 and that of the displacement were normal to the plane of the 

 wave, the force would be resolvable into two, one perpen- 

 dicular to the plane of the wave, which (by the principle of 

 transversal vibrations) can produce no effect ; and the other 

 in the direction of the displacement itself, which will thus 

 be communicated from particle to particle without change. 



