THE ELECTRIC AND LUMINIEEROUS MEDIUM. 
249 
1 (J'p g'Q -}- -j- const. ; and when the medium extends continuously to a 
distance from the seat of the electric action, the constant in this expression must be 
null. When the medium is isotropic, the translatory force is all, there being no 
torque on the element of volume. In a fluid dielectric this compensating hydrostatic 
pressure actually exists, and has been measured ; in a solid it is merely a compendious 
expression for the material reaction per unit volume against the electric forces 
transmitted by the ssther from other matter at a distance. If however the fluid 
dielectric is heterogeneous there wall not be a potential function, and it can only be in 
equilibrium when stratified in a certain manner; if gravity did not operate the 
surfaces of stratification would be the equipotentials of the field of force. 
36. When there are in the electric field interfaces of transition between different 
dielectrics, there will also exist surface-tractions on them which may be evaluated 
by considering an actual, somewhat abrupt, interface to be the limit of a rapid 
continuous variation of the properties of the medium which takes place across a layer 
of finite though insensible thickness. Let then the total displacement {/", g", ), 
Avith its circuital characteristic wdiere there is no free charge, be made up of the 
dielectric material polarization {/', g', li), and the displacement proper [f, g. h) 
which is the sethereal elastic rotation (P, Q, R)/47r. Thus if we neglect now the 
minute difference between the rethereal force (P', Q', R') and the electric force 
(P, Q, R), 
clfidx + dg'Idy -f dh’ldz = - p , d.fjdx + dgjdy + dhldz = p + p', 
where p is the Poisson ideal volume-density corresponding to the polarization, and p 
is the volume-density of free electrons, surface distributions being now by hypothesis 
non-existent."^' The mechanical forcive acting in the dielectric is, per unit volume, 
a force (X', Y', Z') and a torque (L', M', N'), where 
X' =f'dFldx -f g'dFIdy -f h'dF/dz -f pP, L' = g'F - h'Q. 
The component parallel to x of the aggregate force acting on the whole transitional 
layer is the value of jX'hr integrated throughout it. This integral is finite, although 
the volume of integration is small, on account of the large values of the differential 
coefficients which occur in the expression for X. To evaluate it, we endeavour 
* The notation of Part II. is hei’e maintained; thus (/", g", h") represents the (/, g, h) of Maxwell s 
‘T reatise.’ Electrostatic units are here employed. It may be well to recall the relations of these 
quantities. As the asthereal elemental rotation is from its nature circuital, the increment in its outward 
flux across any closed surface is equal to the amount of electrons that have crossed that surface into 
the enclosed region, arising partly from movement of free electrons, and pai'tly from orientation of polar 
molecules over the surface so that one pole is inside and the other outside. Thus, (I, vi, n) being the 
direction vector of the noi'inal, and A representing a finite increment, 
A J (If +mg -p nli) f7S = A | /j (/t — A J {If -f- mg' -p nW) (7S ; 
so that I {If -P mg" -P nh") dS = | /> dr, wdiich gives df'jdx -p dg"ldy -p dh"jdz — />. 
2 K 
VOL. CXC. —A. 
