REPORT ON PHYSICAL OPTICS. 385 



From the construction now alluded to it appears that there 

 are two directions, — the normals, namely, to the two circular 

 sections of the ellipsoid, — in which the velocity of the two rays 

 is the same. These directions are called by Fresnel the ojjtic 

 axes ; although he sometimes applies this term to the normals 

 to the circular sections of the surface of elasticity, or the direc- 

 tions in which a plane ivave is propagated with a single velocity. 

 It thus appears that crystals have in general two optic axes, and 

 can have no more. When tivo of the three principal elasticities 

 are eqiial, the two optic axes unite, and the wave-surface re- 

 solves 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 refrac- 

 tion 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 hiaxal 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 experiments of M. Biot on lunpid 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 ex- 

 periment. This has been done by Fresnel himself by the ordi- 

 nary method of prismatic refraction, as well as by the nicer 

 means afforded by the displacement of the diffracted fringes j 

 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. 



The phenomenon of dispersion, in singly- refracting substances, 

 proves that the elasticity of the vibrating medium varies with 

 the length of the wave. The same thing must take place in 



* Mkm, Inst., torn. iii. 

 1834. 2 c 



