IN PERFECTLY TRANSPARENT MEDIA. 209 



22. The manner in which the ellipsoid (24) may be partially 

 determined by the relations of symmetry which the medium may 

 possess, has been sufficiently discussed in the former paper. 



With respect to the quantity 0, and the surfaces which determine 

 it, the following principle is of fundamental importance. If one body 

 is identical in its internal structure with the image by reflection of 

 another, the values of in corresponding lines in the two bodies will 

 be numerically equal but have opposite signs.* 



It follows that if a body is identical in internal structure with its 

 own image by reflection, the value of (if not zero for all directions) 

 must be positive for some directions and negative for others. More- 

 over, the above described surface by which <J> is represented must 

 consist of two conjugate hyperboloids, of which one is identical in 

 form with the image by reflection of the other. This requires that 

 the hyperboloids shall be right cylinders with conjugate rectangular 

 hyperbolas for bases. A crystal characterized by such properties will 

 belong to the tetragonal system. Since = for the optic axis, it 

 would be difficult to distinguish a case of this kind from an ordinary 

 uniaxial crystal, unless the ellipsoid (24) should approach very closely 

 to a sphere.! 



It is only in the very limited case described in the last paragraph 

 that a medium which is identical in its internal structure with its 

 image by reflection can have the property of circular or elliptic 

 polarization. To media which are unlike their images by reflection, 

 and have the property of circular polarization, we may apply the 

 following general principles. 



If the medium has any axis of symmetry, the ellipsoid or hyper- 

 boloids which represent the values of will have an axis in the same 

 direction. If the medium after a revolution of less than 180 about 

 any axis is equivalent to the medium in its first position, the ellipsoid 

 or hyperboloids will have an axis of revolution in that direction. 



23. The laws of the propagation of light in plane waves, which 



* The necessity of the opposite signs will perhaps appear most readily from the 

 consideration that the direction of rotation of the plane of polarization must be opposite 

 in the two bodies. 



t There is no difficulty in conceiving of the constitution of a body which would have 

 the properties described above. Thus, we may imagine a body with molecules of a 

 spiral form, of which one-half are right-handed and one-half left-handed, and we may 

 suppose that the motion of electricity is opposed by a less resistance within them than 

 without. If the axes of the right-handed molecules are parallel to the axis of X, and 

 those of the left-handed molecules to the axis of Y, their effects would counterbalance 

 one another when the wave-normal is parallel to the axis of Z. But when the wave- 

 normal (of a beam of linearly polarized light) is parallel to the axis of X, the left-handed 

 molecules would produce a left-handed (negative) rotation of the plane of polarization, 

 the right-handed molecules having no effect ; and when the wave-normal is parallel to 

 the axis of Y, the reverse would be the case. 

 G. II. 



