222 MR. G. H. LIVENS ON THE 



This is the general form of the result obtained by SCHWARZSCHILD* that in the 

 special case when we are concerned only with free electrons moving in an sethereal 

 field free from matter their equations of motion can be derived by the variational 

 principle, using the integral 



P dt [LO- Ze0 + 2 - ( Ar)l > 



Jti L c J 



where <j>, A, the ordinary scalar and vector potentials, are regarded as functions of the 

 time and space variables only. 



We now see why it is that consistent formulae have been obtained by different 

 authors using apparently different expressions to represent the field energies. The 

 results are in fact all explicitly independent of any particular interpretations for these 

 energy expressions. 



8. The general dynamical formulation of 6 agrees with the fact that the material 

 media of the field have an internal constitution which enables them to resist the 



setting up of the electric and magnetic polarisations by forces EH [^ OT B] and 



G 



B \r m , E-f E ] respectively, and that in consequence of any change in the polarised 



c 



state of these media their intrinsic energy of elastic or motional type is increased by 

 the amount 



(d 

 where we have used 



E' = E+ l - |V m B], B' = B- I [?,, E 



Thus in setting up the electric field and its associated dielectric polarisations in the 

 medium the potential energy of the field is increased by the amount 



1 f 

 on account of the establishment of the sethereal field together with 



f<fof(E'JP) 



on account of the material polarisation, both amounts being reckoned as potential 

 energy. 



This gives a total for the field equal to 



dv \ j (E SE) + (E P) + - ([r, H B] JP)j- 

 J [4ir c 



* ' 



Gott. Nachr. (Math.-phys. Kl.),' 1903, p. 125. 



