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



In the electrical theory, the kinetic energy is not determined 



by the simple formula of ordinary dynamics from the square 



of the velocity of each element, but is found by integrating 



the product of the velocities of each pair of elements divided 



by the distance between them. Very elementary considerations 



suffice to show that a quantity thus determined when estimated 



per unit of volume will vary as the square of the wave-length. 



h 2 

 We may therefore set F^ 2 — for the kinetic energy, F being 



a constant. The potential energy does not consist in distor- 

 tion of the medium, but depends upon an elastic resistance 

 to the separation of the electricities which constitutes the 

 electrical displacement, and is proportioned to the square of 

 this displacement. The average value of the potential energy 

 per unit of volume will therefore be represented in the elec- 

 trical theory by GA 2 , where G is a constant, and the equation 

 of energies will be 



which gives 



F^ 2 — , = GA 2 (3) 



p* \ i 



V 3 = - = —. (4) 



» 2 F K } 



Both theories give a constant velocity, as is required. But it 

 is instructive to notice the profound difference in the equations 

 of energy from which this result is derived. In the elastic 

 theory the square of the wave-length appears in the potential 

 energy as a divisor; in the electrical theory it appears in the 

 kinetic energy as a factor. 



Let us now consider how these equations will be modified by 

 the presence of ponderable matter, in the most general case of 

 transparent and sensibly homogeneous bodies. This subject is 

 rendered much more simple by the fact that the distances be- 

 tween the ponderable molecules are very small compared with 

 a wave-length. Or, what amounts to the same thing, but may 

 present a more distinct picture to the imagination, the wave- 

 length may be regarded as enormously great in comparison 

 with the distances between neighboring molecules. Whatever 

 view we take of the motions which constitute light, we can 

 hardly suppose them (disturbed as they are by the presence of 

 the ponderable molecules) to be in strictness represented by the 

 equations of wave-motion. Yet in a certain sense a wave- 

 motion may and does exist. If, namely, instead of the actual 

 displacement at any point, we consider the average displace- 

 ment in a space large enough to contain an immense number 

 of molecules, and yet small as measured by a wave-length, such 

 average displacements may be represented by the equations of 



