J. W. Gibbs — Elastic and Electrical Theories of Light. 471 



wave-motion ; and it is only in this sense that any theory of 

 wave-motion can apply to the phenomena of light in trans- 

 parent bodies. When we speak of displacements, amplitudes, 

 velocities (of displacement), etc., it must therefore be under- 

 stood in this way. 



The actual kinetic energy, on either theory, will evidently be 

 greater than that due to the motion thus averaged or smoothed, 

 and to a degree presumably depending on the direction of the 

 displacement. . But since displacement in any direction may 

 be regarded as compounded of displacements in three fixed 

 directions, the additional energy will be a quadratic function of 

 the components of velocity of displacement, or, in other words, 

 a quadratic function of the direction-cosines of the displace- 

 ment multiplied by the square of the amplitude and divided 

 by the square of the period.* This additional energy may be 

 understood as including any part of the kinetic energy of the 

 wave-motion which ma} 7 belong to the ponderable particles. 

 The term to be added to the kinetic energy on the electric 



A 2 

 theory may therefore be written f^ — , where f^ is a quadratic 



P 

 function of the direction-cosines of the displacement. The 

 elastic theory requires a term of precisely the same character, 

 but since the term to which it is to be added is of the same 

 general form, the two may be incorporated in a single term of 



A 2 

 the form A D — , where A D is a quadratic function of the direc- 

 tion-cosines of the displacement. We must, however, notice 

 that both A D and/^ are not entirely independent of the period 

 For the manner in which the flux of the luminiferous medium 

 is distributed among the ponderable molecules will naturally 

 depend somewhat upon the period. The same is true of the 

 degree to which the molecules may be thrown into vibration., 

 But A D and f^ will be independent of the wave-length, (except 

 so far as this is connected with the period,) because the wave- 

 length is enormously great compared with the size of the mole- 

 cules and the distances between them. 



The potential energy on the elastic theory must be increased 

 by a term of the form b v A 2 , where b v is a quadratic function of 

 the direction-cosines of the displacement. For the ponderable 

 particles must oppose a certain elastic resistance to the dis- 

 placement of the ether, which in seolotropic bodies will pre- 

 sumably be different in different directions. The potential 

 energy on the electric theory will be represented by a single 

 term of the same form, say G D A 2 , where a quadratic function 

 of the direction-cosines of the displacement, Gr D , takes the place 



* For proof in extenso of this proposition, when the motions are supposed 

 electrical, the reader is referred to volume xxiii of this Journal, page 268. 



