FRESNEL ON DOUBLE REFRACTION. 299 



different elasticities and two directions of the radius vector which 

 satisfy the condition of a maximum or a minimum. It is easy 

 to perceive, Avithout calculating the double values of («) and of 

 (/3), that these two directions must always be at right angles to 

 each other ; for it results from the general theorem concerning 

 the three rectangular axes of elasticity, that if we consider only 

 the displacements which are performed in one plane and the 

 components comprised in the same plane, not considering the 

 forces which are perpendicular to it, it contains always two rec- 

 tangular directions, for which the resultant of the components 

 comprised in this plane acts along the line of the displacement 

 itself. Now these directions are precisely those which we have 

 just sought ; since, as we have shown, every small displacement 

 parallel to the greatest or least radius vector of any diametral 

 section whatever, excites in the plane of this section a force 

 parallel to the same radius vector, the other component being 

 always perpendicular to this plane. 



Media constituted as we have supposed cannot give more than 

 two images of the same object. 



Hence the tw^o modes of vibration, which are propagated 

 without deviation of their oscillations or change of velocity, are 

 performed in directions at right angles to each other, that is to 

 say, in the most indepeiident manner ; and since, besides, there 

 are only two values of (l»^) or of the elasticity which they put in 

 play, there can be only two systems of waves parallel to the plane 

 of the incident wave, whatever be the original direction of the 

 vibratory motion, since it can always be decomposed along these 

 two directions. If therefore a crystal constituted as we suppose 

 the vibrating medium to be, that is so that the axes of elasticity 

 are parallel throughout its whole extent, be formed into a prism, 

 there can never be seen but two images of a very distant point 

 of sight. The same is also true when this point is so near to 

 the crystal as to render it necessary to take into account the 

 curvature of the wave. 



In fact, it results from the principle of the path of quickest 

 arrival, and from the construction deduced from this by Huy- 

 gens for determining the direction of the refracted ray, that the 

 number of images is equal to the number of points of contact of 

 the tangent planes, which can be drawn on the same side through 

 a straight line to the surfaces of the different waves into which 



