290 FRESNEL ON DOUBLE REFRACTION. 



say, for the tangent of the angle which this plane makes with 

 the axis of x, show that there are two planes equally inclined to 

 the plane of x y, which satisfy the condition of cutting the sur- 

 face of elasticity in a circle, and that there are only these two 

 planes. Every other diametral section has therefore two unequal 

 axes ; so that the waves which are parallel to it ma)' traverse the 

 same medium with two different velocities, according as their 

 vibrations have the direction of one or the other of these axes. 



The Double Refraction becomes nothing for Waves parallel to the 

 two circular sections of the Surface of Elasticity. 

 On the contrary, waves parallel to the circular sections must 

 always have the same velocity of propagation in whatever direc- 

 tion their vibrations be performed, since the radii vectores of 

 each section are all equal to each other ; and, moreover, their 

 vibrations cannot undergo any deviation in passing from one 

 stratum to another, because the component perpendicular to 

 each of these radii vectores is at the same time perpendicular to 

 the plane of the circular section ; for we have demonstrated by 

 the preceding calculations that this condition was fulfilled when 

 the differential of the radius vector became equal to zero. Now 

 this is what takes place for all the radii vectores of the circular 

 sections, since their length is a constant quantity. Consequently, 

 if a crystal be cut parallel to each of the circular sections of the 

 surface of elasticity, and if we introduce into it perpendicularly 

 to these faces rays polariicd in any azimuth whatever, they will 

 not undergo in the crystal either double refraction or deviation 

 of their plane of polarization. Hence these two directions will 

 possess the properties of what have been improperly called the 

 axes of the crystal, and which I shall name optic axes, to distin- 

 guish them from the three rectangular axes of elasticity, which 

 ought, in my opinion, to be considered as the true axes of the 

 doubly-refracting medium. 



There are never more than two optic axes in refracting media 

 whose axes of elasticity have everywhere the same direction. 

 A remarkable consequence of the calculation which we have 

 made is, that a body constituted as we suppose it to be, that is 

 whose particles are arranged in such a manner that the axes of 

 elasticity for each point of the vibrating medium are parallel 

 throughout its whole extent, cannot have more than two oj)tic 



