192 DOUBLE REFRACTION AND THE DISPERSION OF 



Since S, T, and T' are essentially positive quantities, their ratios 

 must be positive. Equation (21) therefore requires that the index 

 of refraction shall increase as the period or wave-length in vacuo 

 diminishes. Experiment has shown no exceptions to this rule, except 

 such as are manifestly attributable to the absorption of light. 



14. It remains to consider the relations between the optical pro- 

 perties of a medium and the planes or axes of symmetry which it 

 may possess. If we consider the statical energy per unit of volume 

 (S) and the period as constant, we may regard equation (2) as the 

 equation of an ellipsoid, the radii vectores of which represent in 

 direction and magnitude the amplitudes of systems of weaves having 

 the same statical energy. In like manner, if we consider the kinetic 

 energy of the irregular part of the flux per unit of volume (T') and 

 the period as constant, we may regard equation (9) as the equation 

 of an ellipsoid, the radii vectores of which represent in direction and 

 magnitude the amplitudes of systems of waves having the same 

 kinetic energy due to the irregular part of the flux. These ellipsoids, 

 which we may distinguish as the ellipsoids (A, B, etc.) and (A', B', etc.), 

 as well as the ellipsoid before described, which we may call the 

 ellipsoid (a, &, etc.), must be independent in their form and their 

 orientation of the directions of the axes of coordinates, being deter- 

 mined entirely by the nature of the medium and the period of 

 oscillation. They must therefore possess the same kind of symmetry 

 as the internal structure of the medium. 



If the medium is symmetrical about a certain axis, each ellipsoid 

 must have an axis parallel to that. If the medium is symmetrical 

 with respect to a certain plane, each ellipsoid must have an axis at 

 right angles to that plane. If the medium after a revolution of less 

 than 180 about a certain axis is then equivalent to the medium in 

 its first position, or symmetrical with it with respect to a plane at 

 right angles to that axis, each ellipsoid must have an axis of revolution 

 parallel to that axis. These relations must be the same for light of 

 all colors, and also for all temperatures of the medium. 



15. From these principles we may infer the optical characteristics 

 of the different crystallographic systems. 



In crystals of the isometric system, as in amorphous bodies, the 

 three ellipsoids reduce to spheres. Such media are optically isotropic 

 at least so far as any properties are concerned which come within the 

 scope of this paper. 



In crystals of the tetragonal or hexagonal systems, the thrt 

 ellipsoids will have axes of rotation parallel to the principal crystal- 

 lographic axis. Since the ellipsoid (a, b, etc.) has but one circular 

 section, there will be but one optic axis, which will have a fixed 

 direction. 



