Page:ComstockInertia.djvu/6

Since the proof of equations (1) is equally valid for relative motion, the integrals involving ($v_{2}$) in the above correspond to flow of energy with respect to axes moving with the system. There is also an implicitly involved internal term in each of the Poynting vector integrals. Since the system is isolated, the sum of such " internal " terms must on the average vanish. There remains therefore to represent the actual average rate of transfer of energy through space only the explicit Poynting terms and the terms involving ($v_{1}$).

The electromagnetic momentum corresponding to any electrical system is given by the components

${\begin{cases}M_{x}={\frac {1}{4\pi V}}\int (\gamma Y-\beta Z)d\tau \\M_{y}={\frac {1}{4\pi V}}\int (\alpha Z-\gamma X)d\tau \\M_{z}={\frac {1}{4\pi V}}\int (\beta X-\alpha Y)d\tau \end{cases}}$

(4)

which, except for the factor $V^{2}$, are the same as the integrals of the components of the Poynting vector throughout the system. Hence equation (3) may be written

$Wv_{1}=lM_{x}+mM_{y}+nM_{z}-lv_{1}\int (lX_{x}+mY_{x}+nZ_{x})d\tau \,$ $-mv_{1}\int (lX_{y}+mY_{y}+nZ_{y})d\tau -nv_{1}\int (lX_{z}+mY_{z}+nZ_{z})d\tau \,$

(3A)

Also if the electrical system here dealt with is to represent a material body, we may assume that the resultant momentum ($M$) is in the direction of the velocity, and hence

$M=lM_{x}+mM_{y}+nM_{z}\,$.

This may be considered as due to the fact that the lack of symmetry necessarily involved in the intimate structure of any electromagnetic system has become a symmetrical average in particles large enough to be dealt with. This symmetrical point may of course have been reached in the case of single atoms. We may now write (3A) in the form

${\begin{array}{l}Wv_{1}=V^{2}M-v_{1}\int \left\{\left(l^{2}X_{x}+m^{2}Y_{y}+n^{2}Z_{z}\right)\right.\\\\\qquad \left.+lm(X_{y}+Y_{x})+ln(X_{z}+Z_{x})+mn(Y_{z}+Z_{y})\right\}d\tau \end{array}}$

(5)

7. To reduce this expression further requires some relation to be established between the stresses and the electric and magnetic force intensities. This process is closely analogous to the derivation of the Maxwell stress in the free aether