COLORS IN PERFECTLY TRANSPARENT MEDIA. 187 



to any point in such a plane. In a wave-crest (or plane in which 

 has a maximum value), Pot will also have a maximum value, 

 which we may call K. For intermediate points we may determine 

 its value from the consideration that the total disturbance may be 

 resolved into two systems of waves, one having a wave-crest, and the 

 other a nodal plane passing through the point for which the potential 

 is sought. The maximum amplitudes of these component systems 

 will be to the maximum amplitude of the original system as 



11 Hi 



cos27Ty and sin2?rj to unity. But the second of the component 



systems will contribute nothing to the value of the potential. We 

 thus obtain 



47T 2 U 47T* 



Comparing this with equation (6), we have 



"-( (7) 



Hence, and by equations (4), 



The kinetic energy of the regular part of the flux is therefore, for 

 each unit of volume, 



. (8) 



i 



9. With respect to the kinetic energy of the irregular part of the 

 flux, it is to be observed that, since ', tf y f ' have their average values 

 zero in spaces which are very small in comparison with a wave-length, 

 the integrations implied in the notations Pot ', Pot if, Pot f ' may be 

 confined to a sphere of a radius which is small in comparison with a 

 wave-length. Since within such a sphere ', rf, f ' are sensibly deter- 

 mined by the values of r\, at the center of the sphere, which is the 

 point for which the value of the potentials are sought, Pot ', Pot jy', 

 Pot f ' must be functions evidently linear functions of r\, f ; and 

 $ Pot *, rf Pot tf, f ' Pot f ' must be quadratic functions of the same 

 quantities. But these functions will vary with the position of the 

 point considered with reference to the adjacent molecules. 



Now the expression for the kinetic energy of the irregular part 

 of the flux, 



