270 J. W. Gibbs—Double Refraction and the 
the direction of vibration, must hold true not only of su 
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. e 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 infinitesimally 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 con- 
straint, while the wave-length remains constant, the period will 
be stationary. Therefore, if the direction of the unconstrained 
vibration is infinitesimally varied by constraint, and the period 
remains rigorously constant, the wave-length will be stationary. 
ence, if we make a central section of the above describe 
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 funda- 
mental 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 to derive from the same equations the laws 
of the dispersion of colors, we shall not be able to obtain an 
equally definite result, since the quantities A, B, etc., and 
A’, B’, etc., are unknown functions of the period. If, however, 
experiment. 
Pr we set 
qa Ae t+ BA +Oy* + Eby +Fya+Gah (16) 
: p 
* See Rayleigh’s Theory of Sound, vol. i, p. 84. 
