288 FRESNEL ON DOUBLE REFRACTION. 



B (a2 _ c') cos X cos Z - C {a^ - 6^) cos X cos Y 

 + (6"^ - c^) cos Y cos Z = 0, 

 a relation similar to that of equation (A.), which determines the 

 direction of the axes of the diametral section, as may be easily- 

 seen by effecting the multiplications. Therefore the directions 

 of these two axes do in reality possess the property announced ; 

 whence it results that the parallel vibrations preserving always 

 the same direction, have a velocity of propagation proportional 

 to the square root of the elasticity put into play, a velocity which 

 may then be represented by the radius vector (o). 



Determination of the Velocity of Propagation of plane and 

 indefinite Waves. 



By the aid of this principle and of the equation of the surface 

 of elasticity, whenever the three semi-axes a, b, c are known, it 

 will be easy to determine the velocity of propagation of plane 

 and indefinite waves whose direction is given. To this end, in 

 the first place, a plane parallel to the waves is to be drawn 

 through the centre of the surface of elasticity, and their vibra- 

 tory motion decomposed into two others in the directions of the 

 greatest and least axis of this diametral section. If we denote 

 by («) the angle made by the incident vibrations with the former 

 of these axes, cos a and sin « will represent the relative intensi- 

 ties of the two components ; and their velocities of propagation, 

 measured perpendicularly to the waves, will be respectively equal 

 to half of the semi-axis of the diametral section to which the 

 vibrations are parallel. These two semi-axes being in general 

 imequal, the two systems of waves will traverse the medium with 

 different velocities, and will cease to be parallel on emerging 

 from the refracting medium if the surface of emergence is oblique 

 to that of the waves, so that the difference of velocities causes a 

 difference of refraction. With regard to the planes of polariza- 

 tion of the two divergent beams, they wiU be perpendicular to 

 each other, since their vibrations are at right angles to each 

 other. 



There are two diametral planes which cut the Surface of Elasticity 

 in circles. 



It is to be remarked that the surface 



u^ = a^ cos^ X + b". cos^ Y + ^2 . cos^ Z, 



