FRESNEL ON DOUBLE REFRACTION. 285 



dicular to the radius vector ; the former is represented in mag- 

 nitude by the square of the length of this radius vector itself; 

 the second, not being perpendicular to the plane of the diame- 

 tral section except for two particular positions, may be gene- 

 rally decomposed into two other forces, one comprised in this 

 plane and the other normal to the plane ; this latter, as we have 

 said, exerts no influence on the propagation of the luminous 

 waves ; but it is not so for the other component, which must be 

 combined with the first component parallel to the radius vector, 

 to obtain the whole elastic force excited in the plane of the 

 waves. 



It will be remarked that in this general case, the elastic force 

 which propagates the waves will not be parallel to the displace- 

 ments which have produced it; whence would result, in the 

 vibrations which pass from one stratum to another, a gradual 

 change of their direction, and consequently of the intensity of 

 the elastic force which they put in play, which would render 

 very difficult the calculation of their propagation, and would 

 prevent the ajiplication to it of the ordinary law, according to 

 which the velocity of propagation is proportional to the square 

 root of the elasticity put in play, a law whose applicability we 

 have shown only for the particular case where the direction of 

 the vibrations and the elasticity remain constant from one stra- 

 tum to another. 



But there exist always in each plane two rectangular direc- 

 tions, such that the elastic forces excited by displacements parallel 

 to each of them being decomposed into two other forces, one 

 parallel, the other perpendicular to this direction, the second 

 component is found perpendicular to the plane ; so that the vi- 

 brations are propagated solely by an elastic force parallel to the 

 primitive displacements, which therefore preserves the same 

 direction and the same intensity during their transit. Now 

 whatever be the direction of the incident vibrations, they may 

 always be decomposed along these two rectangular directions in 

 the diametral plane parallel to the waves, and thus reduce the 

 problem of their path to the calculation of the velocities of pro- 

 pagation of vibrations parallel to these two directions ; a calcu- 

 lation easy to make according to the principle that the velocities 

 of propagation are proportional to the square roots of the elasti- 

 cities put into play, which jirinciplc then becomes rigorously 

 aj)plicable. 



