282 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
integral is then taken to represent the actual distribution of the organized energy in 
the medium when in. electric equilibrium, and not merely its total amount: and 
variation of it with respect to the material configuration should on that hypothesis 
give the actual bodily distribution of mechanical forcive, not merely its statical 
resultant on the hypothesis that the system is absolutely rigid. Now in finding the 
variation of W arising from a virtual displacement {Sx, Si/, Sz) of the polarized 
material, we have to respect the conditions that the free charge pSr is merely 
displaced, so that by the equation of continuity Sp + d [pSx)jdx + d {pSi/)jdy + 
d {pSz)ldz = 0, and also that each element of the material is moved on wdth its own 
K, so that SK + dKjdx .Sx + dKjdy. Sy + dK/dz .Sz — 0 ; while things have been 
arranged so that a variation of V produces no result,—but only however no aggregate 
result on integration by parts. Unless the transitions at interfaces are supposed to 
be gradual, and the integration then to e.xtend throughout all space, there will also 
be direct surface terms in the variation, because the virtual shift of the material leaves 
a space unoccupied on one side and occupies a new space on the other; thus finally 
by the ordinary process of integration by parts we obtain for an}^ region 
SW = 
(lY 1 dK/dV^- dV2 dV 
P ^ + 0 _ , 7 .. (^..2 + ■: 7„2 + 
dx Stt dx \ dx^ 
K /W 
-f 
df 
dV^- 
dz- 
8a: + 
8r 
j Stt V did 
d^\ 
dz^ j 
-b ^^ 
The coefficient of Sx with sign changed has been taken to be the component of the 
bodily mechanical forcive. But to obtain the total mechanical forcive acting on an 
element we must retain all the terms in the variation that belono- to it, so that it 
is illegitimate in this connexion to transmit a traction from it to the boundary of the 
medium by the process of integration by parts. If then we consider the single 
element of volume by itself, so that in the formula 8S is an element of its surface, 
the forcive on it will be von Helmholtz’s one together with a hydrostatic pressure 
— K/Stt, {dY'jdx" fi- dN'^jdy" + dY'^/'dz-) acting over its surface; and this complete 
specification would agree with our previous results, except that we have K — J in 
place of K for reasons already assigned. But it would seem that the method thus 
described must be radically unsound; it would be valid if there -were only one 
medium under consideration, of which W is the energy function : but there is here, 
in the same space, the aether with its stress and the polarized matter with its 
reacting mechanical forces, and (§ G) there is no means of disentangling from a single 
energy function in this way the portions of energy which are associated with these 
different effects. 
66. There are also subsidiary terms in yon Helmholtz’s formulae, involving the 
rate of alteration of the inductive capacity of the ffuid dielectric bv^ compression, 
terms which are extended in the work of Korteweg, Loebejrg, and Kirchhoff to 
include the alterations of the inductive capacity of a solid dielectric produced by the 
