THE ELECTRIC AND LU:\[INrFEROLTS MEDIUM. 
285 
C!<yii>tevv<ition of Energy in the Electric Field: Eiiiiitecl Vcdiditij o/’P oynting’s 
Principle. 
69. It has been explained (§ 6) that the agencies in an electric field may be in part 
traced by transmission through the Eetlier after the manner of ordinary mechanical 
stress, and in part, namely as regards forces on the electrons, not so traced. As 
legaids the former part, therefore, the increase of energy in any region must be 
ex]3iessible explicitly as a surface integral, representing work done by tractions 
exerted over its boundary. This theorem will thus have application in all cases in 
which the configuration of the electrons is not changing ; for its strict application, the 
bodies inside the region which carry currents or electric charges or are polarized, 
must thus be at rest, and there must be no change in the state of electrification of 
any conductor in the region. Eecurring for an illustration to the simpler circum¬ 
stances of a perfect fluid containing vortex rings, it is easy to show analytically that 
the rate of increase of energy in any region is there exj)ressible as a surface integral, 
in\olving the velocity and the pressure, only when thei'e are no vortex rings in the 
region or when the rings in it are all supposed to be held fixed by constraint.^ This 
illustration also emphasizes the point that the surface integral must be taken as a 
whole, that an element of it does not necessarily represent the activity across the 
corresponding portion of the surface. 
Thus taking the material system, concerning which we need make no hypothesis as 
regards inductive quality or jeolotropy, to be at rest in the electric field so that there 
are no changes of energy due to the mechanical forcives, and neglecting those due to 
convection currents which rearrange electrifications, if W and T denote the organized 
potential and kinetic electric energies in the region, and D the rate of dissipation of 
organized energy due to currents of conduction, dFN/dtdl jdtD must be 
expressible as a surface integral. Now W is made up of the energy of eethereal 
strain (Sn) Q"-}-Pt^) c?r, and that of material polarization, l<f) dr where 
(f> = dj'+ Qdg' -f Hdli) which must be an exact differential when there is no 
dielectric hysteresis; thus in all dW/dt = 1{P df'/dt + Q dg"/dtUdE'/dt) dr. 
Again the rate of dissipation arising from ionic migration in the conducting circuits 
^ df /dt) + Q (r — dg”jdt) -{- R (yv — dh”/dt)] dr. Hence we must 
have dTjdt = clEjclt — \{Pu + Qu + dr, in which dPIdt is a surface integral ; 
and this equation will give an a priori indication, independent of special hypothesis, 
of the distribution of organized kinetic energy in the medium, that of the potential 
The reason is simply that the form of the contained vortex ring's is not a function merely of the 
state of the fluid inside the surface, but also in part determines the simultaneous velocity of the fluid 
thioughout all space. So also the energy associated vrith each atom of matter is really distributed 
throughout the whole oether, and therefore the energy-changes associated with changes in the configura¬ 
tion of matter cannot be represented as propagated step by step across the fether. 
