226 ELASTIC AND ELECTRICAL THEORIES OF LIGHT. 



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

 between two nodal planes, the velocities of displacement will be as 



7 O 



fL iL 



- , and the kinetic energy will be represented by A% , where A is a 



constant depending on the density of the medium. The potential 

 energy, which consists in distortion of the medium, may be represented 



h 2 

 by Bjg, where B is a constant depending on the rigidity of the 



medium. The equation of energies, on the elastic theory, is therefore 



I 2 B 

 which gives ^ 2== ~2 == T > (^) 



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 



h 2 

 square of the wave-length. We may therefore set F^ 2 -^ for the kinetic 



energy, F being a constant. The potential energy does not consist 

 in distortion 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 electrical theory by Gh 2 , where G is a 

 constant, and the equation of energies will be 



(3) 

 p 



which gives 



72 ft 



V 2 



"~" 



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 between the ponderable 

 molecules are very small compared with a wave-length. Or, what 



