Since $\sqrt{-g}$~becomes multiplied by~$\lambda^{4}$ on gauge-transformation we must combine it with co-invariants which become multiplied by~$\lambda^{-4}$. The following are easily seen to be in-invariant-densities: \begin{gather*} (\Star{G})^{2} \sqrt{-g};\quad \Star{G}_{\mu\nu}\, \Star{G}^{\mu\nu} \sqrt{-g};\quad \Star{B}_{\mu\nu\sigma}^{\epsilon}\, \Star{B}_{\epsilon}^{\mu\nu\sigma} \sqrt{-g}, \Tag{(88.1)} \\ F_{\mu\nu} F^{\mu\nu} \sqrt{-g}. \Tag{(88.2)} \end{gather*} We can also form in-invariant-densities from the fundamental tensor of the sixth rank. Let $\Star{}(\Star{B}_{\mu\nu\sigma\rho})_{\alpha\beta}$ be the second co-covariant derivative of the co-tensor $\Star{B}_{\mu\nu\sigma\rho}$; the spur formed by raising three suffixes and contracting will vary as~$\lambda^{-4}$ and give an in-invariant-density on multiplication by~$\sqrt{-g}$. \index{Density!in-invariant-}% There are three different spurs, according to the pairing of the suffixes, but I believe that there are relations between them so that they give only one independent expression. The simplest of them is \[ g^{\mu\nu} g^{\sigma\rho} g^{\alpha\beta}\, \Star(\Star{B}_{\mu\nu\sigma\rho})_{\alpha\beta} \sqrt{-g} = \Star{\,\Wave}\, \Star{G} · \!\sqrt{-g}. \Tag{(88.3)} \] If $\mf{A}$~stands for any in-invariant-density, \index{Absolute change!properties of a region}% \[ \int \mf{A}\, d\tau \] taken over a four-dimensional region is a pure number independent of coordinate-system and gauge-system. Such a number denotes a property of the region which is absolute in the widest sense of the word; and it seems likely that one or more of these numerical invariants of the region must stand in a simple relation to all the physical quantities which measure the more general properties of the world. The simplest operation which we can perform on a regional invariant appears to be that of Hamiltonian differentiation, and a particular importance will therefore be attached to the tensors $\Ham A/\Ham g_{\mu\nu}$, $\Ham A/\Ham\kappa_{\mu}$. It has been pointed out by Weyl that it is only in a four-dimensional world that a simple set of regional in-invariants of this kind exists. In an odd number of dimensions there are none; in two dimensions there is one,~$\Star{G} \sqrt{-g}$; in six or eight dimensions the in-invariants are all very complex \PageSep{206} involving derivatives of at least the fourth order or else obviously artificial. This may give some sort of reason for the four dimensions of the world. The \index{Dimensions, world of $3 + 1$!reason for four}% \index{Four dimensions of world}% argument appears to be that a world with an odd number of dimensions could contain nothing absolute, which would be unthinkable.