The curl of~$\kappa_{\mu}$ has an important property; if \[ F_{\mu\nu} = \frac{\dd\kappa_{\mu}}{\dd x_{\nu}} - \frac{\dd\kappa_{\nu}}{\dd x_{\mu}}, \] we see by~\Eq{(85.52)} that \[ F_{\mu\nu}' = F_{\mu\nu}, \Tag{(85.6)} \] so that $F_{\mu\nu}$~is independent of the gauge-system. This is only true of the covariant tensor; if we raise one or both suffixes the function~$\lambda$ is introduced by~\Eq{(85.43)}. It will be seen that the geometry of the continuum now involves $14$~functions which vary from point to point, viz.\ ten~$g_{\mu\nu}$ and four~$\kappa_{\mu}$. These may be subjected to transformations, viz.\ the transformations of gauge discussed above, and the transformations of coordinates discussed in Chapter~\ChapNum{II}\@. Such transformations will not alter any intrinsic properties of the world; but any changes in the~$g_{\mu\nu}$ and~$\kappa_{\mu}$ other than gauge or coordinate transformations will alter the intrinsic state of the world and may reasonably be expected to change its physical manifestations. The question then arises, How will the change manifest itself physically if we alter the~$\kappa_{\mu}$? All the phenomena of mechanics have been traced to the~$g_{\mu\nu}$, so that presumably the change is not shown in mechanics, or at least the primary effect is not mechanical. We are left with the domain of electromagnetism which is not expressible in terms of $g_{\mu\nu}$~alone; and the suggestion arises that an alteration of~$\kappa_{\mu}$ may appear physically as an alteration of the electromagnetic field. We have seen that the electromagnetic field is described by a vector already called~$\kappa_{\mu}$, and it is an obvious step to identify this with the $\kappa_{\mu}$ introduced in Weyl's geometry. According to observation the physical condition of the world is not completely defined by the~$g_{\mu\nu}$ and an additional vector must be specified; according to theoretical geometry the nature of a continuum is not completely indicated by the~$g_{\mu\nu}$ and an additional vector must be specified. The conclusion is irresistible that the two vectors are to be identified. Moreover according to~\Eq{(85.52)} we can change $\kappa_{\mu}$ to $\kappa_{\mu} + \dd\phi/\dd x_{\mu}$ by a change of gauge without altering the intrinsic state of the world. It was explained at \PageSep{202} the beginning of \SecRef{74} that we can make the same change of the electromagnetic potential without altering the resulting electromagnetic field. We accordingly accept this identification. The $\kappa_{\mu}$~and $F_{\mu\nu}$ of the present geometrical theory will be the electromagnetic potential and force of Chapter~\ChapNum{VI}\@. It will be best to suspend the convention $\kappa_{\mu}^{\mu} = 0$ \Eq{(74.1)} for the present, since that would commit us prematurely to a particular gauge-system.