FRESNEL ON DOUBLE REFRACTION. 317 



therefore the difference between the two values of (/), or the 

 quantity sought, is equal to 



(_/ — h) sin n . sin m ; 

 consequently this difference is proportional to the product of the 

 sines of the two angles {m) and {n) ; which was to be proved. 



The angles concerned are those which the common direction 

 of the ordinary and extraordinary rays makes with the two dia- 

 meters of the ellipsoid perpendicular to the circular sections, 

 which diameters we have called optic axes, admitting that this 

 name ought to be given to the two directions along which the 

 luminous rays traverse the crystal without undergoing in it any 

 double refraction. But it is to be remarked that in general these 

 rays meet the element of the surface of the luminous waves to 

 which they correspond obliquely. Now we have previously 

 pointed out, that if the surface of the crystal were parallel to this 

 element or to its tangent plane, the normal direction would be 

 that which must be given to the incident beam, in order that it 

 might not undergo double refraction in penetrating into the 

 crystal ; whence it would appear that we ought also to give the 

 name of optic axes to these two directions of the incident rays 

 which do not coincide with the two normals to the circular sec- 

 tions of the ellipsoid. Hence the direction of the optic axes 

 Avould be different according as we determined it by the direction 

 of the incident rays, perpendicular at the same time to the sur- 

 face of the incident waves and the refracted waves, or by the 

 direction of the refracted rays corresponding to these waves. In 

 truth, this difference is very slight in almost all crystals with two 

 axes ; but there are some of them in which it becomes more per- 

 ceptible, and where the two directions can no longer be con- 

 founded. That to which it appears most fitting to give the name 

 of optic axis of the crystal is the direction of the refracted rays 

 which traverse it without undergoing double refraction. Adopt- 

 ing this definition, the law of the product of the sines of the 

 angles which any ray makes with the two optic axes, becomes a 

 rigorous consequence of our theory, as we have just proved. 



Hitherto we have occupied ourselves solely with the velocity 

 and the direction of the waves and rays : we now proceed to in- 

 vestigate their planes of polarization. 



