FRESNEL 0\ DOUBLE REFRACTION. 307 



tions of its axes are given by observation, and will probably 

 afford in each crystal a very simple relation with its lines of cry- 

 stallization and its faces of cleavage*; two of these axes divide 

 into two equal parts the acute and obtuse angles comprised be- 

 tween the two optical axes, the direction of which may be deter- 

 mined immediately by observation, and the third axis of elasticity 

 is perpendicular to the plane of the two optical axes. 



The directions of the axes of elasticity may also be found by 

 observing those of the planes of polarization of the emergent 

 light by the aid of a very simple rule relative to these planes de- 

 duced by M. Biot from his experiments, and which is found to 

 be a consequence of our theory, as we shall soon demonstratef. 

 As to the constants a, b, c, or the three semi-axes of the surface 

 of elasticity, they represent by hypothesis the velocities of pro- 

 pagation of vibrations parallel to the axes of ,«?, y and ss, that is 

 to say, the spaces which they describe during the unit of time. 

 These velocities may be determined in several ways. The most 

 direct is to measure successively the velocities of the rays refracted 

 parallel to each of the axes of elasticity, and whose vibrations are 

 parallel to one of the other two axes. For this purpose may be 

 employed the ordinary observations of refraction, or the more 

 delicate process furnished by the principle of interferences, and 

 which allows of the most minute differences of velocity being 

 estimated. In traversing the crystal parallel to the axis of x, the 

 light assumes two velocities, which being measured give [b] and 

 (c) ; parallel to the axis of y these two velocities are [a) and (c), 

 and parallel to the axis of 5" they are {a) and {b). Hence two of 

 these measurements, made with care, are, rigorously speaking, 

 sufficient to determine the three quantities a, b and c. 



• It would seem that the axes of elasticity should always assume directions 

 symmetrical with regard to the corresponding faces of the crystal, that is to say, 

 that they should be axes of symmetry for the form, as they are for the elasticity ; 

 yet M. Mitsclierlich has observed several crystals in which the line which divides 

 into two equal parts the angle of the two optical axes, is not found directed 

 symmetrically with regard to the corresponding faces of crystallization. 



+ In s lying that the simple and elegant construction given by M. Biot for 

 determining the planes of polarization is a consequence of oiu' theory, I do not 

 mean it to he understood tliat I have any right to ))articipate in the honour of 

 this discovery, since the labours of M. Biot on double refraction are much ear- 

 lier than mine. I mean simply to state that the law which he had found flows 

 necessarily from the theory which I have set forth ; and that we have here a 

 striking confirmation, and not merely a fact which, by the help of an arbitrary 

 constant or by the addition of a subsidiary hypothesis, is made to coincide with 

 the calculation. 



