272 J. W. Gibbs— Double Refraction and the 
tion. Our equations will apply to such a constrained type of 
oscillation, and A, B, ete., and A’, B’, etc., and therefore H and 
H’, will have the same values in the last described system of 
waves as in the first system, although the wave-length and the 
period have been varied. Therefore, in differentiating equation 
18), which is essentially an equation between 7 and p, or its 
equivalent (19), we may treat H and H’ as constant. This 
gives 
We thus obtain the values of H’ and H: 
H= 17__27k* 2mk*X dn 
— si— 2 3 dv’ 
a. 
as Qnn® “aN (20) 
By determining the values of H and H’ for different directions 
of oscillation, we may determine the values of A, B, etc., and 
', B’, ete 
By means of these equations, the ratios of the statical energy 
(S), the kinetic energy due to the regular part of the flux (T), 
and the kinetic energy due to the irregular part of the flux (T’), 
are easily obtained in a form which admits of experimental de- 
termination. Equations (8) and (9) give 
2 Lf 
se let Se ra. 
Therefore, by (20), 
T’ 27H’ 27H'n? Adn_ dilogn (21) 
a ee Si Ae EO A. 
Bote oy ae A—dlogn_dlogi (22) 
x T : dloga diog A 
‘aes _ dog n (23) 
Ss dlogl 
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 excep- 
tions to this rule, except such as are manifestly attributable to 
the,absorption of light. : 
14. It remains to consider the relations between the optical 
properties 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 regar 
equation (2) 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 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, 
