320 FEESNEL ON DOUBLE REFRACTION. 



is perpendicular to the luminous vibrations, I shall now prove 

 that the planes of polarization of the ordinary and extraordinary- 

 waves divide into two equal parts the dihedral angles formed by 

 the two planes drawn along the normal to the wave, and the two 

 normals to the planes of the circular sections of the surface of 

 elasticity. 



The rule given by M. Biot for determining the direction of the 

 planes of polarization of the ordinary and extraordinary rays 

 agrees with the theory set forth in this memoir. 

 Suppose, in fact, that this surface be cut by a diametral plane 

 parallel to the wave, the two axes of this section will give the 

 directions of the ordinary and extraordinary vibrations ; if then 

 we di'aw through the centre two planes perpendicular to these 

 two diameters, these will be the planes of polarization respect- 

 ively of the ordinary and extraordinary vibrations. Now it must 

 be remarked, — 1st, that they will each pass through one of the 

 axes of the section, since these latter are perpendicular to each 

 other ; 2nd, that the axes of the diametral section cutting it each 

 into two symmetrical portions, must divide into equal parts the 

 acute and obtuse angles formed by the two lines along which 

 the plane of this section meets those of the circular sections, 

 since in these two directions the radii vectores of the diametral 

 section are equal to each other, as belonging at the same time to 

 two circular sections which have the same diameter. 



This being established, conceive a sphere concentric with the 

 surface of elasticity ; the plane of the diametral section, and the 

 two planes of the circular sections, will trace on this sphere a 

 spherical triangle, of which the side contained in the first plane 

 will be divided into two equal parts by one of the planes of 

 polarization ; its supplementary triangle will be that formed by 

 the normals of these three planes drawn through the common 

 centre, that is to say, which will result from the intersection of 

 the spherical surface with the three planes drawn along these 

 three normals taken two and two. Now the planes which divide 

 into two equal parts the sides of the first triangle also divide 

 into two equal parts the angles of the second ; this is an easily 

 proved property of supplementary triangles. Therefore the 

 plane of polarization, which divides into two equal parts the 

 side of the first triangle comprised in the diametral section, 

 divides also into two equal parts the corresponding angle of the 



