118 REPORT ON PHYSICAL OPTICS. 



are two directions the normals, namely, to the two circular sec- 

 tions of the ellipsoid, in which the velocity of the two rays is 

 the same. These directions are called by Fresnel the optic axes r 

 although he sometimes applies this term to the normals to the 

 circular sections of the surface of elasticity, or the directions in 

 which a plane wave is propagated with a single velocity. It thus 

 appears that crystals have in general two optic axes, and can have 

 no more. When two of the three principal elasticities are equal, 

 the two optic axes unite, and the wave-surface resolves itself into 

 the sphere and spheroid of revolution. Thus the form of the wave 

 in uniaxal crystals, which Huygens assumed as the most natural, 

 comes out as a simple corollary from the general theory of FresneL 

 When, lastly, the three elasticities are all equal, the wave-surface 

 becomes a sphere; the velocity is accordingly the same in all 

 directions, and the law of refraction is reduced to the known law 

 of Snellius. 



It was easily shown to follow from the general construction, 

 that the difference of the squares of the reciprocal velocities of the 

 two rays, in biaxal crystals, is proportional to the product of the 

 sines of the angles which their common direction within the crystal 

 contains with the two axes; so that the remarkable law of Sir 

 David Brewster and M. Biot is brought under the same theory. 

 But it appeared further, from that theory, that the velocity of 

 neither of the rays is constant, and that the refraction of both is 

 performed according to a new law. This conclusion was at variance 

 with all the received notions upon the subject ; and indeed the ex- 

 periments of M. Biot on limpid topaz* seemed to warrant his- 

 assumption that the refraction of one of the rays followed the 

 ordinary law of the sines. It became, therefore, a matter of much 

 interest to decide this question by accurate experiment. This has 

 been done by Fresnel himself by the ordinary method of prismatic 

 refraction, as well as by the nicer means afforded by the dis- 

 placement of the diffracted fringes ; and the result in both cases 

 has been conclusive in favour of his theory. The numerical data 

 afforded by the observations of M. Biot on topaz enabled Fresnel 

 to compute, according to the principles of that theory, the velocity 

 of the ray in different directions ; and the observed variation was- 

 found to agree with that deduced. 



* Mem. Iml., torn. iii. 



