328 FRESNEL ON DOUBLE REFRACTION. 



suppress the factors c"- {a + b) and a~ [b — c) without sensibly 

 altering the equation, so long as [a) does not differ much from 

 (c), that is to say, so long as the double refraction has not a very 

 great energy. 



When toe have given the angle between the two optic axes, it is 

 sufficient to know two of three constants, a, b, c, in order to 

 determine the third. 



It is sufficient to know [a) and (c), that is, the greatest and 

 least velocity of light in the crystal, together with the angle be- 

 tween the two optic axes, to determine the other semi-axis {b), 



c /a^ — b^ 

 since the tangent of half this angle is equal to — \/ .^ _ 3 , 



a known function of three quantities, a, b and c. It was by 

 pursuing this method that I calculated, with the elements of 

 double refraction given by M. Biot for topaz, the variations of 

 velocity which the ordinary beam must undergo in it, before 

 seeking to verify them by experiment, and I found them very 

 nearly such as the calculation had given me. The theory also 

 pointed out to me in what direction the ordinary beam had the 

 most different velocities. 



For topaz it is the smallest axis of the surface of elasticity or 

 of the ellipsoid which divides into equal parts the acute angle of 

 the two optic axes, and the two limits of the velocities of the 

 ordinary ray are (a) and [b) ; now the ordinary beam has the 

 velocity («) when it is parallel to the axis of {y), since («) is the 

 greatest radius vector of the perpendicular diametral section 

 made in the ellipsoid, and since the corresponding plane of 

 polarization, that is perpendicular to the radius vector [a], is 

 also that of the ordinary beam, as passing within the acute angle 

 of the two optic axes. The velocity of this same beam becomes 

 equal to (i) when the light traverses the crystal parallel to the 

 axis of {x), because then the diametral plane perpendicular to 

 this direction cuts the ellipsoid in an ellipse whose greatest 

 radius vector is (6). Moreover, the plane perpendicular to {b), 

 or the corresponding plane of polarization, belongs to the ordi- 

 nary refraction ; for it is also contained in the acute angle formed 

 by the two planes drawn along the luminous ray and each of the 

 optic axes, a dihedral angle which then becomes equal to zero, 

 these two planes becoming coincident with that of the two optic 



