THE ELECTKIC AND LUMINIFEROUS MEDIUM. 
225 
so that is only in part an electrostatic potential. Inside a uniform isotropic 
conductor at rest, the condition of circuitality becomes = dpjdt ] substituting 
this, we have dpjdt -f- IttC'^ctK ^p = 0, so that p — p^ exp (— 47 rc”crK“h), showing that 
an initial volume density of free electricity would in that case be instantly driven 
to the boundary owing to the dielectric action. This proposition may be extended 
to seolotropic media. 
14. The nature of the foregoing electric scheme may be elucidated by aid of some 
simple applications. 
(I.) A/Vhen a conducting system is in steady motion so that there is no conduction 
current flowing into it, the electric force (P, Q, R) must be null throughout its 
substance. Thus for the case of a solid conductor rotating round an axis of 
symmetry in a uniform magnetic field parallel to that axis, with steady angular 
velocity oi, the electric force in it, namely {oicx — cW^jdx, cocy — d'^Jdy, — d-^^jdz), 
must be null, so that T'l = wc + y^) + A; the polarization in it is therefore 
null, but there is in it an sethereal displacement — {4:Ttc^)~^ [djdx, djdy, djdz)'^^. 
In outside space, the electric force and sethereal force are each — (djdx, djdy, djdz) 
Avhere "'Fj is that free electrostatic potential which is continuous with the surface 
value ^ (t)C {x^ + + A at the conductor. Inside the conductor this purely 
sethereal displacement involves an electrification of volume density p = — oicj^Trd^, 
which will be a density of free electrons or ions as all true electrifications are; while 
there is a compensating surface density cr equal to the difference of the total normal 
electric displacements on the two sides, that is to ( 47 rc^)“i {d^Jdn^ + d^r^ldn^), 
wh^e dn^, dn^ are both measured towards the surface, the outside medium being 
air for which K is unity. The value of the constant A is determined by the circum¬ 
stance that the aggregate of this volume and surface charge shall be null when the 
conductor is insulated and unelectrified, or equal to the given total charge when It is 
insulated and charged : when it is uninsulated, the constant is determined by the 
position of the point on it that is connected to Earth, and therefore at zero potential. 
The piocedure of Part II., § 25 is thus justified, because there is in fact no dielectric 
polarization in the conductor, but only aethereal displacement. 
It remains to consider whether the parts of this volume density p and surface 
density cr of electrification are carried round with the conductor in its motion, or slip 
back through its volume and over its surface so as to maintain fixed positions in 
space. It is clear (as in Part II., § 27) that the same cause, namely, viscous diffusion 
of momentum among moving ions and molecules, which produces Ohmic resistance 
to a steady current, will lead to the electrons constituting electric densities being 
wholly carried on by the matter whenever a steady state is attained. This necessary 
consequence of the theory is in keeping with Rowland’s classical experiments on 
convection currents. The excessively minute magnetic field due to these convection 
currents themselves has been neglected in the above analysis, which has enabled us 
to specify the slight redistribution of free charge on the rotating conductor when 
VOL. CXC.—A. 2 G 
