ELASTIC AND ELECTRICAL THEORIES OF LIGHT. 225 



supposition.* It should be added that all diversity of opinion on 

 this subject has been confined to those whose theories are based on 

 the dynamics of elastic bodies. Defenders of the electrical theory 

 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 quadratic 

 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 something 

 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 may 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 system 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 



* "At the same time, if the above reasoning be valid, the question as to the direction 

 of the vibrations in polarized light is decided in accordance with the view of Fresnel. 

 ... I confess I cannot see any room for doubt as to the result it leads to. ... I only 

 mean that if light, as is generally supposed, consists of transversal vibrations similar to 

 those which take place in an elastic solid, the vibration must be normal to the plane of 

 polarization." Lord Rayleigh " On the Light from the Sky, its Polarization and Color;" 

 Phil. Mag. (4), xli (1871), p. 109. 



" Green's dynamics of polarization by reflexion, and Stokes' dynamics of the diffraction 

 of polarized light, and Stokes' and Rayleigh's dynamics of the blue sky, all agree in, as 

 it seems to me, irrefragably, demonstrating Fresnel's original conclusion, that in plane 

 polarized light the line of vibration is perpendicular to the plane of polarization." 

 Sir Wm. Thomson, loc. citat. 



fThe terms kinetic energy and potential energy will be used in this paper to denote 

 these average values. 



G. II. p 





