THE ELECTRIC AND LUMINIFEROUS MEDIUM. 
255 
the local part of the regular electric force acting at a molecule has already been 
assigned (§ 19) as "sttI', very approximately for the case of fluid media, possibly not 
so approximately for solids. The argument was that owing to the translational 
mobility of the surrounding molecules, their action on the one under consideration 
averages into that of the uncompensated distribution of poles which would exist on 
the surface of a small spherical cavity in a continuous uniformly polarized medium,— 
or, more precisely, into that of a spherical shell of poles, of thickness not indefinitely 
small but with this law of distribution around the centre. For the interior of a 
uniformly polarized medium the local part of the electric force is thus at each instant 
constant throughout this cavity and equal to ^iri '; therefore the mechanical force 
exerted on the polar molecule (that is one involving equal numbers of positive and 
negative electrons) at the centre of the cavity is null, as it depends on the rate of 
variation of this electric force. But at a place where the polarization varies from 
point to point, the alteration in the law of surface-density over the cavity will supply 
a local part. 
When the polarization i' changes onl}^ in magnitude and not in direction, this part 
will arise from a distribution of uncompensated poles over the surface of the cavity, ol 
density — -f a; di'Jdx -f y di'Jdy + 5; di'Jdz) cos 6, where the subsciipt zero 
implies the value at the centre. If the axis of x is taken along the direction of i' 
the electric potential U in the interior due to this distribution is equal to 
— VTT ( t 
I X 0 
3 dx 
X 
— f 73- 
di. 
di\ 
dy 
0 
dz 
xz 
On a molecule of moment /r,„ at the centre, this gives a force 
— ix,4/dx {djdx, didy, djdz) U, that is -|7r/a., {^djdx, djdy, d/dz) i 
0- 
Thus there is a bodily force due to this cause, of intensity |77 {^dldx, djdy, djdz) i'Q^ ; 
but there is not any bodily torque. 
42. Now let us proceed to the general case, in which the direction ol the 
polarization {J', y', /d), as well as its magnitude, varies Irom point to point; in the 
hypothetical case in which the effective distance between the poles ol a molecule is 
small compared with the average distance between neighbouring molecules, we 
can express the molecular part of the foreive on an element of volume by simple 
summation for f, g', and li separately, by aid of the expressions just found. 
Thus it consists of a bodily force (X^, Y^, Z^) and torque (Lj^, M^, N^), where 
Xi = fd?Jdx + g'dFJdy + h'dFJdx, = g'li^ - h'Q,, U,) being the local 
part of the electric force in the spherical cavity, so that 
dh' 
'dx 
Hence 
