J. W. Gibbs — Elastic and Electrical Theories of Light. 469 



have always placed the electrical displacement at right angles 

 to the " plane of polarization." It will, however, be better to 

 assume this direction of the displacement as probable rather 

 than as absolutely certain, not so much because many are likely 

 to entertain serious doubts on the subject, as in order not to 

 exclude views which have at least a historical interest. 



The wave-velocity, then, for any constant period, is a quad- 

 ratic function of the cosines of a certain direction, which is 

 probably that of the displacement, but in any case determined 

 by the displacement and the wave-normal. The coefficients of 

 this quadratic function are functions of the period of vibration. 

 It is important to notice that these coefficients vary separately, 

 and often quite differently, with the period, and that the case 

 does not at all resemble that of a quadratic function of the 

 direction- cosines multiplied by a quantity depending on the 

 period. 



In discussing the dynamics of the subject we may gain some- 

 thing in simplicity by considering a system of stationary waves, 

 such as results from two similar systems of progressive waves 

 moving in opposite directions. In such a system the energy 

 is alternately entirely kinetic and entirely potential. Since the 

 total energy is constant, we mav set the average kinetic energy 

 per unit of volume at the moment when there is no potential 

 energy equal to the average potential energy per unit of 

 volume when there is no kinetic energy.* We may call this 

 the equation of energies. It will contain the quantities I and 

 p, and thus furnish an expression for the velocity of either sys- 

 tem of progressive waves. We have to see whether the elastic 

 or the electric theory gives the expression most conformed to 

 the facts. 



Let us first apply the elastic theory to the case of the so- 

 called vacuum. If we write h for the amplitude measured in 

 the middle between two nodal planes, the velocities of dis- 

 placement will be as — , and the kinetic energy will be rep- 

 resented by A—, where A is a constant depending on the den- 



P 

 sity of the medium. The potential energy, which consists in 



h 2 

 distortion of the medium, may be represented by B— , where 



B is a constant depending on the rigidity of the medium. The 

 equation of energies, on the elastic theory, is therefore 



A-. = B- (!) 



.72 ~T> 



which gives V 2 = —= = — . (2) 



p* A v ' 



* The terms kinetic energy and potential energy will be used in this paper to 

 denote these average values. 



