Take a displacement at~$P$ (coordinates,~$x_{\mu}$) and transfer it by parallel displacement to an infinitely near point~$P'$ (coordinates, $x_{\mu} + dx_{\mu}$). Let its initial length measured by the gauge at~$P$ be~$l$, and its final length measured by the gauge at~$P'$ be~$l + dl$. We may express the change of length by the formula \[ d(\log l) = \kappa_{\mu}\, dx_{\mu}, \Tag{(85.1)} \] where $\kappa_{\mu}$~represents some vector-field. If we alter the gauge-system we shall, of course, obtain different values of~$l$, and therefore of~$\kappa_{\mu}$. It is not necessary to specify the route of transfer for the small distance $P$ to~$P'$. The difference in the results obtained by taking different routes is by~\Eq{(84.4)} proportional to the area enclosed by the routes, and is thus of the second order in~$dx_{\mu}$. As $PP'$~is taken infinitely small this ambiguity becomes negligible compared with the first-order expression~$\kappa_{\mu}\, dx_{\mu}$. Our system of reference can now be varied in two ways---by change of coordinates and by change of gauge-system. The behaviour of~$g_{\mu\nu}$ and~$\kappa_{\mu}$ for transformation of coordinates has been fully studied; we have to examine how they will be transformed by a transformation of gauge. A new gauge-system will be obtained by altering the length of the standard at each point in the ratio~$\lambda$, where $\lambda$~is an arbitrary function of the coordinates. If the standard is decreased in the ratio~$\lambda$, the length of a displacement will be increased in the ratio~$\lambda$. If accents refer to the new system \[ ds' = \lambda\, ds. \Tag{(85.2)} \] The components $dx_{\mu}$ of a displacement will not be changed, since we are not altering the coordinate-system, thus \[ dx_{\mu} = dx_{\mu}'. \Tag{(85.3)} \] Hence \[ g_{\mu\nu}'\, dx_{\mu}'\, dx_{\nu}' = ds'^{2} = \lambda^{2}\, ds^{2} = \lambda^{2} g_{\mu\nu}\, dx_{\mu}\, dx_{\nu} = \lambda^{2} g_{\mu\nu}\, dx_{\mu}'\, dx_{\nu}', \] so that \[ g_{\mu\nu}' = \lambda^{2} g_{\mu\nu}. \Tag{(85.41)} \] %\PageSep{201} It follows at once that \index{Potential!electromagnetic}% \begin{align*} g' & = \lambda^{8} g, \Tag{(85.42)} \\ {g'}^{\mu\nu} &= \lambda^{-2} g^{\mu\nu}, \Tag{(85.43)} \\ \sqrt{-g'} · d\tau' &= \lambda^{4} \sqrt{-g} · d\tau. \Tag{(85.44)} \end{align*} Again, by~\Eq{(85.1)} \begin{align*} \kappa_{\mu}'\, dx_{\mu} &= d(\log l') = d\{\log(\lambda l)\} \\ &= d(\log l) + d(\log\lambda) \\ &= \kappa_{\mu}\, dx_{\mu} + \frac{\dd(\log\lambda)}{\dd x_{\mu}}\, dx_{\mu}. \end{align*} Or, writing \[ \phi = \log\lambda, \Tag{(85.51)} \] then \[ \kappa_{\mu}' = \kappa_{\mu} + \frac{\dd\phi}{\dd x_{\mu}}. \Tag{(85.52)} \]