THE ELECTRIC AND LUMINIFEROUS MEDIU^M. 
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voluinB cl mechanical foicive the same as the one thus deduced, from the energy 
function. An infinite number of such stress-systems might in fact be specified, for 
there are six components of stress which need satisfy only three conditions. If 
however the stress system is required to be symmetrical with respect to the lines of 
polaiization, theie is in this respect no indefiniteness 40) j and the one given by 
AON Helmholtz is of this kind, Thus the definite result really deduced by 
VON Helmholtz from his energy hypothesis is an expression for the bodily 
mechanical forcive in tlie polarized medium, the (X, Y, Z) of equation (4) of his 
inemoii ; while correlative formulm are applied by him and by Kirchhoff for the 
bodily forcive in a magnetized medium. These expressions, however, definitely 
contradict the formulas of Maxwell and of the other previous writers for the bodily 
mechanical forcive in a magnetized medium, which are in general agreement with 
those developed in this paper : in fact von Helmholtz translates his formulae into 
Maxaa^ell’s electric stress system, while Maxaa^ell himself had to invent for the 
case of magnetic polarization, which Avas the one he considered, a different stress^ 
system, namely his magnetic stress. As recent Avriters have in the main tacitiv 
accepted von Helaiholtz’s procedure, it is incumbent on us to assign the origin of 
tins discrepancy; and for this purpose a summary of his method is given, the effect 
of alteration of the coefficient of polarization arising from strain in the material being 
for the present left out of account. 
65. Ihe organized electric energy in the polarized medium lieing assumed, from 
other considerations, to be 
W = |K/87r . (riV7cZ.r2 + cmjdif -f clT^ldz^) dr, 
Avhere 
djdx (K dY/dx) + djdy (K dY/dy) + djdz (K dYIdz) + 47 rp = 0, 
in Avliich p is a constant associated with the element of dielectric matter, called 
the density of its free electric charge, the forces acting will be derived from 
the variation of W; variation with respect to V leads to the electric forces, 
and that with respect to the material configuration leads to the mechanical 
ones. ihe problem is to determine the mechanical forces when there is electric 
equilibrium, that is Avhen variation with respect to Y yields a null result. The form 
of W above expressed does not lead to this null result ; we can however by 
iiitep’ation by parts derive the form W = -ijVpc^T, the essence of this transfor¬ 
mation being that in the new integral the distribution of the energy among the 
elements of volume dr of the medium has been altered. This form does not satisfy 
the above requirement either, but by combining the two forms Ave obtain 
W = |{pY - K/87r. (dV2/r/x3 + dY'^jdif^ -{- dY'^jdz^^ dr, 
whose variation Avith respect to V is null as required : although as integration by 
parts is employed, tlie variation is not null for each single element of mass. This 
a^ol. CXC.—A. 2 o 
