COLOKS IN PERFECTLY TRANSPARENT MEDIA. 185 



easily appear that the manner in which the general or average flux 

 in any element Dv (represented by TI, f ) distributes itself among 

 the molecules and intermolecular spaces must be entirely determined 

 by the amount and direction of that flux and its period of oscillation. 

 Hence, and on account of the superposable character of the motions 

 which we are considering, we may conclude that the values of ', rf, f 

 at any given point in the medium are capable of expression as linear 

 functions of , q, f in a manner which shall be independent of the 

 time and of the orientation of the wave-planes and the distance of 

 a nodal plane from the point considered, so long as the period of 

 oscillation remains the same. But a change in the period may 

 presumably affect the relation between ', rf, f ' and 77, f to a certain 

 extent. And the relation between ', r\ ', f and r\, f will vary 

 rapidly as we pass from one point to another within the element Dv. 



5. In the motion which we are considering there occur alternately 

 instants of no velocity and instants of no displacement. The statical 

 energy of the medium at an instant of no velocity must be equal to 

 its kinetic energy at an instant of no displacement. Let us examine 

 each of these quantities, and consider the equation which expresses 

 their equality. 



6. Since in every part of an element T>v the irregular as well as the 

 regular part of the displacement is entirely determined (for light of 

 a given period) by the values of q, the statical energy of the 

 element must be a quadratic function of rj, say 



where A, B, etc. depend only on the nature of the medium and the 

 period of oscillation. At an instant of no velocity, when 



sin27r-=0. and cos 2 27r- = l, 



P , P 



the above expression will reduce by equations (1) to 



ni 



(Aa 2 + B/3 2 + Cy 2 + E/3y + Fya + Ga/3)cos 2 27T j D<v. 



Since the average value of cos 2 2 Try in an indefinitely extended 



I/ 



space is J, we have for the statical energy in a unit of volume 



S = J(Aa 2 + B/3 2 + Cy 2 + E^y + Fya + Ga/3). (2) 



7. The kinetic energy of the whole medium is represented by the 

 double volume-integral* 



r 



* The fluxes are supposed to be measured by the electromagnetic system of units. It 

 is to be observed that the difference of opinion which has prevailed with respect to the 

 estimation of the energy of electrical currents does not extend to such as are solenoidal, 

 which may be regarded as composed of closed circuits. 



