Energy in the Electrodynamic Field. 387 



results are obtained. In addition Macdonald's own form 

 and an alternative are deduced, and although reasons are 

 given for their rejection in favour of the simpler and more 

 general type of theory first obtained, the former (Mac- 

 donald's own form) is retained throughout the succeeding 

 discussion for purpose of illustration. The various theories 

 are then subjected to critical examination on general grounds 

 and subsequently in reference to their behaviour in certain 

 special fields, and it is shown that in the present state of our 

 knowledge any alternative to Poynting's theory must be 

 rejected as being wholly incompatible with our physical 

 conceptions of energy and its streaming in the electro- 

 magnetic field. 



2. The general theory of energy streaming is best 

 approached in the tentative manner suggested by Prof. 

 Larmor *, by considering the conservation of energy in any 

 portion of an electromagnetic field. We assume generally 

 that the diminution of energy inside any closed surface in 

 the field is equal to the flow of energy outwards across the 

 surface. In other words, if U represents the total available 

 electrodynamic energy inside the closed surface f, then 

 — dTJ/dt is equal to the flux of energy outwards over the 

 surface, or to the integral 



1 



Sndf 



taken over it : S„ denotes the outward normal component of 

 the vector S determining the energy flux. 



The energy U represents the electrodynamic energy 

 available in the field. If V represents the potential energy 

 and T the kinetic energy, then 



dV dV dT ^ 



dt dt^dt ' 



where F represents the rate of working of the electrical 

 forces converting purely electrical energy into energy of 

 other types. Now if the volume integrals are supposed 

 taken throughout the field inside/, we know that 



8-7T , 

 %J v 



%J V 



where Ci represents the part of the total current of Maxwell's 



* Phil. Trans. 1897, p. 285. 

 2E 2 



