ELASTIC AND ELECTRICAL THEORIES OF LIGHT. 227 



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 displacement 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 wave-motion; and it is only in this sense that 

 any theory of wave-motion can apply to the phenomena of light in 

 transparent bodies. When we speak of displacements, amplitudes, 

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

 ment. 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 displacement 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 may belong to the ponderable particles. The 

 term to be added to the kinetic energy on the electric theory may 



h 2 

 therefore be written /D-|, where / D is a quadratic 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 incor- 



h 2 

 porated in a single term of the form A D ~2, where A D is a quadratic 



function of the direction-cosines of the displacement. We must, 

 however, notice that both A D and / D 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 

 /D will be independent of the wave-length (except so far as this is 



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

 the reader is referred to page 187 of this volume. 



