io The Double Refraction of Light in a Crystallized 



portional to the semidiameters, the elastic forces will be both 

 parallel and proportional to the normals; for the normal, ter- 

 minated by any of the principal planes, is inversely as the per- 

 pendicular on the tangent plane. 



3. If the elastic force be resolved into two, one parallel and 

 the other perpendicular to the direction of the displacement, the 

 former will be inversely as the square of the semidiameter in 

 the direction of the displacement. 



4. If the ellipsoid be cut by a plane through its centre, and 

 if the elastic force arising from a displacement in the direction 

 of either axis of the section be resolved parallel and perpendi- 

 cular to that axis, the part perpendicular to the axis will also be 

 perpendicular to the plane of the section. For (by Lem. 4) the 

 plane passing through one axis and the perpendicular to the 

 tangent plane at its extremity, is perpendicular to the other axis, 

 and therefore to the plane of the section. But if the displace- 

 ment be in the direction of any other diameter of the section, 

 the elastic force, resolved perpendicularly to that diameter, will 

 be oblique to the plane of the section. 



To apply these things to the double refraction of light 

 in a crystallized medium, imagine the ellipsoid to be described 

 as above, and let it be cut through its centre by a plane 

 parallel to that of a plane wave of light incident on the crystal ; 

 then if the vibrations of the light be parallel to either of the 

 axes of the section, the plane containing the direction of the 

 vibrations and that of the elastic force arising from them will 

 be perpendicular to the plane of the wave (No. 4, preceding) ; 

 and therefore, according to Fresnel's theory, the direction of the 

 vibrations will remain parallel to itself, while the wave is pro- 

 pagated. But if the light be common light, or if it be polarized, 

 and the plane of polarization be not perpendicular to either of 

 the axes of the section, the wave will be divided into two others 

 having the directions of their vibrations parallel to the semiaxes 

 of the elliptic section, and their planes of polarization perpendi- 

 cular to them : their velocities of propagation measured in a 

 direction perpendicular to their plane will be different, and 



