Hence, when b* = c*, or b* = a 2 , these angles become 0, or 90 ; 

 and the two optic axes unite> coinciding in the former case, 

 with the axis of greatest elasticity, and in the latter with that 

 of the least. y u~/^^ / ^^ 1 -' 



In -eaeh of these cases, then, o>= a/, and the equations of 

 the wave-surface become 



v~* = cr 2 sin 2 (t) + c~ 2 cos 2 cu, v' = c ; 



the former of which is the equation of the ellipsoid of revolu- 

 tion, and the latter that of the sphere. Accordingly, the 

 wave-surface resolves itself into the sphere and spheroid of the 

 Huygenian law ; and the form of the wave in uniaxal crystals, 

 assumed by Huygens, is deduced as a simple corollary from 

 the general theory of Fresnel. 



Finally, when the three elasticities are all equal, it will 

 appear at once from the preceding equations that the spheroid 

 becomes a sphere. The velocity is accordingly the same in 

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

 known law of Snell. 



(218) It has been stated (94) that, as soon as a class of 

 double-refracting substances was discovered, possessing two 

 optic axes, the construction of Huygens was found not to be 

 general. It was still supposed, however, that the velocity of 

 one of the rays in every crystal was constant ; or, in other 

 words, that one of the rays was refracted according to the 

 ordinary law of the sines. According to Fresnel's theory, 

 however, the velocity of neither of the rays in biaxal crystals 

 was constant, and the refraction of both was performed ac- 

 cording to a new law. It was, therefore, a matter of much 

 interest to decide this question by accurate experiment. 

 This experimental problem was solved by Fresnel himself, 

 and the result was decisive in favour of his theory. 



