These conclusions are somewhat modified by the existence of a particularly simple regional in-invariant, which seems to have been generally overlooked because it is not of the type which investigators have generally studied. The quantity \[ \int \Chg{\surd\{-|\Star{G}^{\mu\nu}|\}}{\sqrt{-|\Star{G}^{\mu\nu}|}}\, d\tau \Tag{(88.4)} \] is an invariant by~\Eq{(81.1)} and it contains nothing which depends on the gauge. It is not \emph{more} irrational than the other in-invariants since these contain~$\sqrt{-g}$. We shall find later that it is closely analogous to the metrical volume and the electromagnetic volume (\SecRef{81}) of the region. It will be \index{Volume!generalised}% called the \emph{generalised volume}. This in-invariant would still exist if the world \index{Generalised volume}% had an odd number of dimensions. It may be remarked that $F^{\mu\nu} \sqrt{-g}$, or~$\mf{F}^{\mu\nu}$, is an in-tensor-density. Thus the factor~$\sqrt{-g}$ should always be associated with the contravariant tensor, if the formulae are to have their full physical significance. The electromagnetic action-density should be written \[ F_{\mu\nu} \mf{F}^{\mu\nu}, \] and the energy-density \[ -F_{\mu\nu} \mf{F}^{\nu\alpha} + \tfrac{1}{4} g_{\mu}^{\nu} F_{\alpha\beta} \mf{F}^{\alpha\beta}. \] The field is thus characterised by an \emph{intensity~$F_{\mu\nu}$} or a \emph{quantity} of density~$\mf{F}^{\mu\nu}$; both descriptions are then independent of the gauge-system used. \Section{89.}{The natural gauge} \index{Natural coordinates!gauge}% For the most part the laws of mechanics investigated in Chapters \ChapNum{III}--\ChapNum{V} have been expressed by tensor equations but not in-tensor equations. Hence they can only hold when a particular gauge-system is used, and will cease to be true if a transformation of gauge-system is made. The gauge-system for which our previous work is valid (if it is valid) is called the \emph{natural gauge}; it stands in somewhat the same position with respect to a general gauge as Galilean coordinates stand with respect to general coordinates.