284 FRESNEL ON DOUBLE REFRACTION. 



propagation ; for this reason we shall now confine ourselves to 

 the determination of the component parallel to the displace- 

 ments. 



The angles which this direction makes with the axes are 

 X, Y, Z ; the cosines of the angles which the same axes make 

 with the resultant are 



o^.cosX i^.cosY c^.cosZ 



~7~' ~7^' ~7~' 



consequently the cosine of the angle which this resultant makes 

 with the direction of displacement is equal to 



g^cos^X + 6^ . cos^Y + c^ . cos^Z 

 / 

 Now this cosine must be multiplied by the force /to obtain its 

 component parallel to this direction ; the component sought is 

 therefore equal to 



a^ . cos2 X + ^2 . cos^ Y + c2 . cos^ Z. 

 If we denote by v^ this ©omponent of the elastic force, in order 

 that the corresponding velocity of propagation may be repre- 

 sented by V, we shall have 



u2 = c^ cos2 X + ^>^ cos2 Y + c^ . cos2 Z. 



Surface of Elasticity, whichrepresents the Law of the Elasticities 

 and of the Velocities of Propagation. 



I shall suppose a surface to be constructed according to this 

 equation, each radius vector of which, making angles equal to 

 X, Y, Z with the axes of x, y, z, has for its length the value of v : 

 we may call it the surface of elasticity, since the squares of its 

 radii vectores will give the components of the elastic force in the 

 direction of each displacement. 



If we conceive a system of luminous waves (always supposed 

 plane and indefinite) which are propagated in the medium whose 

 law of elasticity is represented by this surface, and draw through 

 its centre a plane parallel to the waves, every component per- 

 pendicular to this plane must be considered as having no in- 

 fluence on the velocity of propagation of the luminous waves. 

 The elastic force excited by displacements parallel to one of the 

 radii vectores of this diametral section, may always be decom- 

 posed into two other forces, one parallel and the other perpen- 



