THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
227 
V of tliG piobloni. in the form V'^V — c ^ (^'pcljdx -j- -f- TcljdzY''^, tlie boundary 
condition being that V is constant over each conductor. 
Thus in the case of a system of conductors moving steadily through space witli 
uniform velocity v in the direction of the axis of x, e denoting (1 — we have 
(/ 9, h) = (4770')-^ (P, eQ, ePv), and therefore {d^jdx^ + e d^/dif + e dddz^) V = 0. 
Ihe distribution of electric force is therefore precisely the same as if the system 
were at rest, and the isotropic dielectric constant unity of the surrounding space 
changed into an molotropic one (1, e, e), cf. Part I. §115 ; and so would the surface 
density of true charge, which is the superhcial discontinuity of total displacement, 
be the same, were it not that there is aethereal displacement inside the conductors 
which must be subtracted. The internal displacement current thence arising is 
- vdjdx ( 0 , - VC, vh) ■ hence {a, h, c) is of the form [didx, (1 + 
(1 + uYc')-i djdz] cj), by Ampere’s circuital relation : the circuitality of {a, b, c) then 
leads to a characteristic equation for <^, which must be solved so as to give at the 
surface of the conductor a value for the normal component of (a, h, cj continuous with 
the already known outside value, and the internal displacement is thereby 
deteimined. There is no bodily electrification inside the conductors, since this 
displacement is circuital. 
We can restore the above characteristic equation of V, the potential of the electric 
foice, to an isotropic form by a geometrical strain of the system and the surrounding 
space, represented by (x, y, z') = {^i‘x, y, z) : the actual distribution of potential 
aiound the original system in motion corresponds then to that isotropic distribution 
of potential round the new system at rest which has the same v'alues over the 
conductors. The aethereal displacements through related elements of area 
8S and SS , of direction cosines (/, m, n) and (!', m, n') in the two spaces, multiplied 
by 47rC“, will be 
- {Ididx + e md/dy + e nd/dz) V3S and - {Vdjdx' + mdjdy' + n'djdz') V'SS' ; 
of these the second is always e - times the first; thus the elements of surface for which 
the total displacement is null correspond in the two systems, and therefore the lines 
and tubes of total displacement also correspond, the flux of displacement in these 
tubes being e ' times greater in the second system than in the first. Put on account 
of the aethereal displacement in the interior, the outside tubes do not mark out the 
distilbution of the charge on each conductor. If then a system of charged 
conductors has a velocity of uniform translation v through the aether : and an auxiliary 
system at rest is imagined consisting of the original system and its space each 
uniformly expanded in the ratio d or (1 - in the direction of the motion, 
and the charges on this new system are times those on the actual system ; then 
the fields of aethereal displacement of the two systems agree in the surrounding 
spaces so as to be the same across corresponding areas, but the distributions of the 
chaiges on the conductors do not thus exactly correspond. [These results ai’e 
