258 
HR. J. LARMOR ON A DYNAMICAL THEORY OF 
the clisplciceuient in addition to tliose of the first order, and the ec|Uilibriuin oetween 
two contiguous portions will not depend on continuity of displacement and of surface 
traction alone : other quantities also would have to be continuous for which there is 
no interpretation in the ordinary analysis of elastic reactions : the elastic stress 
would in fact not then be expressible in terms of tractions on interfaces. In such a 
case the only jirocedure that seems open, as the science of mechanics is now con¬ 
stituted, would be to transfer the efi'ects of that part of the elastic eneigy which 
involves higher differential coefiicients to the class of intrinsic or non-mechanical 
deformations. 
45. In this theory of electric polarization the division of the forcive per unit volume 
into a molar and a molecular part lias been made by means of the ideal volume and 
surface densities of Poisson, wdiich are the equivalent as regards outside points of the 
actual polarization of the material. This method consists essentially in computing 
the forcive by combining opposed poles of neighbouring elements, instead ol taking 
the single polarized element as the unit ; it shows that these adjacent poles nearly 
compensate each other except as regards a simple volume density whose attraction 
has no molecular part, and a surface density partly at the outer surface and partly 
at the surface of the cavity which contains the point under consideration. The 
effect of the latter surface density, depending as it does wholly on the immediate 
surroundings, is the molecular or cohesive part of the average forcive. 
These principles may be enforced and illustrated by contrast with a procedure b}^ 
separate molecules which would usually lead to a different result; it will suffice to 
consider the case in which the polarizatioir is uniform in direction throughout the 
material. If the axis of x be taken in the direction in wdiich the intensity of the 
electric polarization changes most rajiidly in the neighbourhood of the point con¬ 
sidered, it is easy to see that the bodily force on an element due to the surrounding 
polar molecules is parallel to x and equal to — {nf'dJ''ldx-{-hg'dg'jdx-\-hh'dh Idx), 
and thus derivable from a potential function — ^ -f hg'^ + tJ/b®), where a, ij are 
constants. In the case of an interface of rapid transition from one uniformly polarized 
medium to another, there is'thus a forcive only in the transition layer, and its integral 
throughout that layer is equivalent to a traction parallel to the axis of .r, of intensity 
siir^ wdiere 0 is the angle between the interface and the axis 
of X, jiulling at the interface into each medium. If the polarization {/', g', /b), or i' , is 
normal to the interface, this traction is — -lab^siiffd, if tangential it is — sin-0. 
To estimate the values of a and !), we may consider separately the forcives exerted by the 
molecules of the polarized medium on /Xy, jx^., the components of a molecular moment 
p situated in the neigbbourhood of the interface, in the case wdien the intertace is 
normal to the axis of a:. The outstanding terms in the aggregate forcive due to the 
surrounding molecules, wdiich do not cancel each other by symmetry, are normal to the 
interface and make up /x.^. -j- d'‘r~^ j dx dij^ -}- ix^^(X:d^r~^/dxdz ~; 
or, per unit volume, “ w d“ wlierein a = — 2b because = 0. 
