186 DOUBLE REFRACTION AND THE DISPERSION OF 



where dv lt dv 2 are two infinitesimal elements of volume, 

 ((+ '\ the corresponding components of flux, r the distance between 

 the elements, and S denotes a summation with respect to the 

 coordinate axes. Separating the integrations, we may write for 

 the same quantity 



It is evident that the integral within the brackets is derived from 

 +' by the same process by which the potential of any mass is 

 derived from its density. If we use the symbol Pot to express this 

 relation, we may write for the kinetic energy 



The operation denoted by this symbol is evidently distributive, 

 so that Pot (+') = Pot + Pot '. The expression for the kinetic 

 energy may therefore be expanded into 



JZ/ Pot gdw + JE/V Pot g dv + JZ/f' Pot gdv + J2/f Pot f dv. 



But (', and therefore Pot', has in every wave-plane the average 

 value zero. Also and therefore Pot has in every wave-plane a 

 constant value. Therefore the second and third integrals in the 

 above expression will vanish, leaving for the kinetic energy 



f2/VPotefo+*2/fPotf<fo, (3) 



which is to be calculated for a time of no displacement, when 



2-Tra u . 27T/3 n U i 2?ry u 



= -- COS27T-7-, y= -cos27r T , f = L cos27r-j-. (4) 

 pi pi pi 



The form of the expression (3) indicates that the kinetic energy 

 consists of two parts, one of which is determined by the regular part 

 of the flux, and the other by the irregular part of the flux. 



8. The value of Pot may be easily found by integration, but 

 perhaps more readily by Poisson's well-known theorem, that if q is 

 any function of position in space (as the density of a certain mass), 



where the direction of the coordinate axes is immaterial, provided 

 that they are rectangular. In applying this to Pot we may place 

 two of the axes in a wave-plane. This will give 



In a nodal plane, Pot^=0, since g has equal positive and negative 

 values in elements of volume symmetrically distributed with respect 



