FRESNEt. ON DOUBLE REFRACTION. 321 



second triangle, that is, the dihedral angle formed by the two 

 planes drawn along the normal to the wave and the diameters 

 perpendicular to the two circular sections ; and for the same 

 reason, the other plane of polarization will divide into two equal 

 parts the supplement of this dihedral angle. 



M. Biot has deduced from his observations on the double re- 

 fraction of topaz and several other bi-axal crystals, the following 

 rule for determining the direction of the planes of polarization 

 of the ordinary and extraordinary rays. 



" Conceive a plane drawn through each of the axes of the cry- 

 stal, and through the ray which undergoes the ordinary refrac- 

 tion. Conceive through this same ray a third plane, which 

 bisects the dihedral angle formed by the two former. The lumi- 

 nous molecules which have undergone the ordinary refraction 

 are polarized in this intermediate plane; and the molecules 

 which have undergone the extraordinary refraction ai'e polarized 

 perpendicularly to the intermediate plane drawn through the ex- 

 traordinary ray according to the same conditions." [Precis EU- 

 mentaire de Physique Experimentale, vol. ii. page 502.) 



The lines which M. Biot here calls tTle axes of the crystal, are 

 those which we have called optic axes. We have remarked, that 

 in order to assimilate in the best manner possible the language 

 of the undulatory system with that of the emission theory, we 

 ought to give the name of optic axis to the direction along which 

 the luminous rays traverse the crystal without undergoing double 

 refraction ; and adopting this definition, we have proved that the 

 law of the product of the two sines is a necessary consequence 

 of our theor}\ The same is not true for the rule of M. Biot 

 relative to the determination of the planes of polarization. His 

 enunciation does not exactly agree with the construction which 

 we have deduced from the properties of the surface of elasticity ; 

 because the dihedral angles bisected by the planes of polarization 

 according to this construction are drawn along the normal to the 

 wave and the two normals to the circular sections of the surface 

 of elasticity, and in general the normal to the wave does not 

 coincide exactly with the direction of the refracted ray, nor the 

 normals to the circular sections of the same surface with the 

 true optic axes, which are the perpendiculars to the circular sec- 

 tions of the ellipsoid. In truth, the geometrical theorem which 

 we have demonstrated for the surface of elasticity applies equally 

 to the ellipsoid ; but the greatest and least radius vector of the 



