FLUXES OF ENERGY IN THE ELECTROMAGNETIC FIELD. 
425 
sistence of energy. But it would be quite inadequate in its older sense referring to 
integral amounts ; the definite localisation by Maxwell, of electric and magnetic 
energy, and of its waste, necessitates the similar localisation of sources of energy ; 
and in the consideration of the supply of energy at certain places, combined with the 
continuous transmission of electrical disturbances, and therefore of the associated 
energy, the idea of a flux of energy through space, and therefore of the continuity of 
energy in space and in time, becomes forced upon us as a simple, useful, and necessary 
principle, which cannot be avoided. 
When energy goes from place to place, it traverses the intermediate space. Only 
by the use of this principle can we safely derive the electromagnetic stress from the 
equations of the field expressing the two laws of circuitation of the electric and 
magnetic forces ; and this again becomes permissible only by the postulation of the 
definite localisation of the electric and magnetic energies. But we need not go so far 
as to assume the objectivity of energy. This is an exceedingly difficult notion, and 
seems to be rendered inadmissible by the mere fact of the relativity of motion, on 
which kinetic energy depends. We cannot, therefore, definitely individualise energy 
in the same way as is done with matter. 
If p be the density of a quantity whose total amount is invariable, and which can 
change its distribution continuously, by actual motion from place to place, its equation 
of continuity is 
couv qp = p, . (1) 
where q is its velocity, and q p the flux of p. That is, the convergence of the flux of 
p equals the rate of increase of its density. Here p may be the density of matter. 
But it does not appear that we can apply the same method of representation to the 
flux of energy. We may, indeed, write 
cony X = T,.(2) 
if X be the flux of energy from all causes, and T the density of localisable energy. 
But the assumption X = Tq would involve the assumption that T moved about like 
matter, with a definite velocity. A part of T may, indeed, do this, viz., when it is 
confined to, and is carried by matter (or ether) ; thus we may write 
conv (qT + X) = T,.(3) 
where T is energy which is simply carried, whilst X is the total flux of energy from 
other sources, and which we cannot symbolise in the form Tq; the energy which 
comes to us from the Sun, for example, or radiated energy. It is, again, often 
impossible to carry out the principle in this form, from a want of knowledge of how 
energy gets to a certain place. This is, for example, particularly evident in the case of 
gravitational energy, the distribution of which, before it is communicated to matter, 
MDCCCXCII.—A. 3 T 
