COLOKS IN PERFECTLY TRANSPARENT MEDIA. 189 



For a given medium and light of a given period, the coefficients a, 6, 

 etc., are constant. 



This relation between the velocity of the waves and the direction of 

 oscillation is capable of a very simple geometrical expression. Let r 

 be the radius vector of the ellipsoid 



l. (14) 



Then 1 _ ax 2 + by 2 + cz 2 + ey z +fzx + gxy 



r 2 r 2 



If this radius is drawn parallel to the electrical oscillations, we shall 



have ; e = y = ? = y 



r p' r p' r p' 

 and V-l. (15) 



That is, the wave-velocity for any particular direction of oscillation 

 is represented in the ellipsoid by the reciprocal of the radius vector 

 which is parallel to that direction. 



11. This relation between the wave-length, the period, and the 

 direction of vibration, must hold true not only of such vibrations as 

 actually occur, but also of such as we may imagine to occur under 

 the influence of constraints determining the direction of vibration in 

 the wave-plane. The directions of the natural or unconstrained 

 vibrations in any wave-plane may be determined by the general 

 mechanical principle that if the type of a natural vibration is infini- 

 tesimally altered by the application of a constraint, the value of the 

 period will be stationary.* Hence, in a system of stationary waves 

 such as we have been considering, if the direction of an unconstrained 

 vibration is infinitesimally varied' in its wave-plane by a constraint 

 while the wave-length remains constant, the period will be stationary. 

 Therefore, if the direction of the unconstrained vibration is infinitesi- 

 mally varied by constraint, and the period remains rigorously constant, 

 the wave-length will be stationary. Hence, if we make a central 

 section of the above described ellipsoid parallel to any wave-plane, 

 the directions of natural vibration for that wave-plane will be parallel 

 to the radii vectores of stationary value in that section, viz., to the 

 axes of the ellipse, when the section is elliptical, or to all radii, when 

 the section is circular. 



12. For light of a single period, our hypothesis has led to a 

 perfectly definite result, our equations expressing the fundamental 

 laws of double refraction as enunciated by Fresnel. But if we ask 

 how the velocity of light varies with the period, that is, if we seek 



* See Rayleigh's Theory of Sound, vol. i, p. 84. 



