When the quadruple integral is regarded as the limit of a sum, the infinitesimal parallelopipeds may be taken of any shape and orientation; but for analytical integration we choose them to be coincident with meshes of the coordinate-system that is being used, viz.\ \[ \delta_{1} x_{\mu} = (dx_{1}, 0, 0, 0);\ \delta_{2} x_{\mu} = (0, dx_{2} , 0, 0);\quad\text{etc.} \] Then \Eq{(49.1)}~reduces to a single diagonal \[ dV = dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}. \] We write $d\tau$ for the volume-element when chosen in this way, so that \[ d\tau = dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}. \] It is not usually necessary to discriminate between~$d\tau$ and the more general expression~$dV$; and we shall usually regard $\sqrt{-g} · d\tau$ as an invariant. Strictly speaking we mean that $\sqrt{-g} · d\tau$ behaves as an invariant in volume-integration; whereas $\sqrt{-g} · dV$ is intrinsically invariant. For Galilean coordinates $x$, $y$, $z$, $t$, we have $\sqrt{-g} = 1$, so that \[ \sqrt{-g}\, d\tau = dx\, dy\, dz\, dt. \Tag{(49.41)} \] Further if we take an observer at rest in this Galilean system, $dx\, dy\,dz$~is his element of proper-volume (three-dimensional)~$dW$, and $dt$~is his proper-time~$ds$. \index{Proper-volume}% Hence \[ \sqrt{-g}\, d\tau = dW\, ds. \Tag{(49.42)} \] By \Eq{(49.41)} we see that $\sqrt{-g}\, d\tau$ is the volume in natural measure of the four-dimensional element. This natural or invariant volume is a physical conception---the result of physical measures made with unconstrained scales; it may be contrasted with the geometrical volume~$dV$ or~$d\tau$, which expresses the number of unit meshes contained in the region. Let $T$~be a scalar, i.e.\ an invariant function of position; then, since $T\sqrt{-g}\, dV$~is an invariant, \[ \int T\sqrt{-g}\, d\tau\quad\text{is an invariant} \] for any absolutely defined four-dimensional region. Each unit mesh (whose edges $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$ are unity) contributes the amount $T\sqrt{-g}$ to this \PageSep{111} invariant. Accordingly we call $T\sqrt{-g}$ the \emph{scalar-density}\footnote {I have usually avoided the superfluous word ``scalar,'' which is less expressive than its synonym ``invariant.'' But it is convenient here in order to avoid confusion between the density \emph{of an invariant} and a density \emph{which is invariant}. The latter,~$\rho_{0}$, has hitherto been called the invariant density (without the hyphen).} or \emph{invariant-density}. \index{Density!scalar-and tensor-}% \index{Scalar-density}% \index{Invariant-density (scalar-density)}%