222 
ME. 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 eethereal 
field free from matter their equations of motion can be derived by the variational 
principle, using the integral 
Lq — 2eof) + 2 i (A?') 
c 
5 
where 0, 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 E + - and 
B—- [r„, E + Eo] respectively, and that in consequence of any change in the polarised 
state of these media their intrinsic energy of elastic or motional type is increased by 
the amount 
jdy j'(E' <5P) + (B'(5I), 
where we have used 
E' = E + i [r,.B], B' = B-1 [f,„ E + EJ. 
c c 
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 
on account of the establishment of the sethereal field together with 
jduj(E'(lP) 
on account of the material polarisation, both amounts feeing reckoned as potential 
energy. 
This gives a total for the field equal to 
j dv j|A (E ffi) + (E JP) + i ([r,.B] JP)| ■ 
* ‘ Giitt. Nachr. (Math.-phys. Kl.),’ 1903, p. 125. 
