Accordingly $m_{g}/m_{e}$~is a constant of nature and it may be absorbed in equation~\Eq{(80.8)} by properly choosing the unit of~$F_{\mu\nu}$. \Section{81.}{Electromagnetic volume} If $a_{\mu\nu}$~is any tensor, the determinant~$|a_{\mu\nu}|$ is transformed according to the law \[ |a_{\mu\nu}| = J^{2} |a_{\mu\nu}'| \] \PageSep{194} by~\Eq{(48.8)}, whence it follows as in~\Eq{(49.3)} that \[ \int \Chg{\surd(|a_{\mu\nu}|)}{\sqrt{|a_{\mu\nu}|}}\, d\tau \Tag{(81.1)} \] for any four-dimensional region is an invariant. We have already considered the case a $a_{\mu\nu} = g_{\mu\nu}$, and it is natural now to consider the case $a_{\mu\nu} = F_{\mu\nu}$. Since the tensor~$g_{\mu\nu}$ defines the metric of space-time, and the corresponding invariant is the metrical volume (natural volume) \index{Electromagnetic action!volume}% \index{Volume!electromagnetic}% of the region, it seems appropriate to call the invariant \[ V_{e} = \int \Chg{\surd(|F_{\mu\nu}|)}{\sqrt{|F_{\mu\nu}|}}\, d\tau \Tag{(81.2)} \] the electromagnetic volume of the region. The resemblance to metrical volume is purely analytical. Since $|F_{\mu\nu}|$~is a skew-symmetric determinant of even order, it is a perfect square, and \Eq{(81.2)}~is rational. It easily reduces to \[ V_{e} = \int (F_{23} F_{14} + F_{31} F_{24} + F_{12} F_{34})\, d\tau. \Tag{(81.31)} \] In Galilean coordinates this becomes \[ V_{e} = \int (\alpha X + \beta Y + \gamma Z)\, d\tau. \Tag{(81.32)} \] It is somewhat curious that the scalar-product of the electric and magnetic forces is of so little importance in the classical theory, for \Eq{(81.32)} would seem to be the most fundamental invariant of the field. Apart from the fact that it vanishes for electromagnetic waves propagated in the absence of any bound electric field (i.e.\ remote from electrons), this invariant seems to have no significant properties. Perhaps it may turn out to have greater importance when the study of electron-structure is more advanced. From~\Eq{(81.31)} we have \begin{align*} V_{e} &= \int \sum \left(\frac{\dd\kappa_{1}}{\dd x_{4}}\, \frac{\dd\kappa_{2}}{\dd x_{3}} - \frac{\dd\kappa_{1}}{\dd x_{3}}\, \frac{\dd\kappa_{2}}{\dd x_{4}}\right) d\tau \intertext{the summation being for all permutations of the suffixes} &= \int \sum \left\{\frac{\dd}{\dd x_{4}} \left(\kappa_{1}\, \frac{\dd\kappa_{2}}{\dd x_{3}}\right) - \frac{\dd}{\dd x_{3}} \left(\kappa_{1}\, \frac{\dd\kappa_{2}}{\dd x_{4}}\right)\right\} d\tau. \end{align*} Hence $V_{e}$~reduces to a surface-integral over the boundary of the region, and it is useless to consider its variations by the Hamiltonian method. The electromagnetic volume of a region is of the nature of a flux through its three-dimensional boundary. \Section{82.}{Macroscopic equations} \index{Macroscopic electromagnetic equations}%