When in Part~II we substitute a conceptual space with still more general geometry, we shall not need to regard it as in opposition to the present discussion. We may learn more from a different graphical picture of what is going on; but we shall not have to abandon anything which we can perceive clearly in the first picture. We consider now the gauging-equation $\Star{G} = 4\lambda$ assumed by Weyl. It is probably the one which most naturally suggests itself. Suppose that we have adopted initially some other gauge in which $\Star{G}$~is not constant. $\Star{G}$~is a co-invariant such that when the measure of interval is changed in the ratio~$\mu$, $\Star{G}$~changes in the ratio~$\mu^{-2}$. Hence we can obtain a new gauge in which $\Star{G}$~becomes constant by transforming the measure of the interval in the ratio~$\Star{G}^{-\frac{1}{2}}$. By~\Eq{(87.6)} the gauging-equation is equivalent to \[ G - 6\kappa_{\alpha}^{\alpha} + 6\kappa_{\alpha} \kappa^{\alpha} = 4\lambda. \Tag{(89.2)} \] But by~\Eq{(54.72)} the proper-density of matter is \begin{align*} \rho_{0} &= \frac{1}{8\pi} (G - 4\lambda) \\ &= \frac{3}{4\pi} (\kappa_{\alpha}^{\alpha} - \kappa_{\alpha} \kappa^{\alpha}). \Tag{(89.3)} \end{align*} For empty space, or for space containing free electromagnetic fields without electrons, $\rho_{0} = 0$, so that \[ \kappa_{\alpha}^{\alpha} = \kappa_{\alpha} \kappa^{\alpha}, \Tag{(89.4)} \] except within an electron. This condition should replace the equation $\kappa_{\alpha}^{\alpha} = 0$ which was formerly introduced in order to make the electromagnetic potential determinate~\Eq{(74.1)}. We cannot conceive of any kind of measurement with clocks, scales, moving particles or light-waves being made \emph{inside} an electron, so that any gauge employed in such a region must be purely theoretical having no significance in terms of practical measurement. For the sake of continuity we define the natural gauge in this region by the same equation $\Star{G} = 4\lambda$; it is as suitable as any other. Inside the electron $\kappa_{\alpha}^{\alpha}$~will not be equal to~$\kappa_{\alpha} \kappa^{\alpha}$ and the difference will determine the mass of the electron in accordance with~\Eq{(89.3)}. But it will be understood that this application of~\Eq{(89.3)} is merely conventional; although it appears to refer to experimental quantities, the conditions are such that it ceases to be possible for the experiments to be made by any conceivable device. \Section{90.}{Weyl's action-principle} \index{Action, material or gravitational!Weyl's formula}% \index{Principle!of least action}% \index{Stationary action, principle of}%