Transforming the $3$-index symbol $[\mu\nu, \sigma]$ by an alteration of gauge we have \index{Christoffel's $3$-index symbols!generalisation of}% \index{Derivative!in-covariant}% \index{In-covariant derivative}% \index{Three-index symbol!generalised}% by~\Eq{(85.41)} \begin{align*} [\mu\nu, \sigma]' &= \frac{1}{2} \left(\frac{\dd(\lambda^{2} g_{\mu\sigma})}{\dd x_{\nu}} + \frac{\dd(\lambda^{2} g_{\nu\sigma})}{\dd x_{\mu}} - \frac{\dd(\lambda^{2} g_{\mu\nu})}{\dd x_{\sigma}}\right)\displaybreak[0] \\ &= \lambda^{2} [\mu\nu, \sigma] + \tfrac{1}{2} g_{\mu\sigma}\, \frac{\dd\lambda^{2}}{\dd x_{\nu}} + \tfrac{1}{2} g_{\nu\sigma}\, \frac{\dd\lambda^{2}}{\dd x_{\mu}} - \tfrac{1}{2} g_{\mu\nu}\, \frac{\dd\lambda^{2}}{\dd x_{\sigma}}\displaybreak[0] \\ &= \lambda^{2} [\mu\nu, \sigma] + \lambda^{2}(g_{\mu\sigma} \phi_{\nu} + g_{\nu\sigma} \phi_{\mu} - g_{\mu\nu} \phi_{\sigma}) \end{align*} by~\Eq{(85.51)}. We have written \[ \phi_{\mu} \equiv \frac{\dd\phi}{\dd x_{\mu}}. \] Multiply through by ${g'}^{\sigma\alpha} = \lambda^{-2} g^{\sigma\alpha}$; we obtain \[ \{\mu\nu, \alpha\}' = \{\mu\nu, \alpha\} + g_{\mu}^{\alpha} \phi_{\nu} + g_{\nu}^{\alpha} \phi_{\mu} - g_{\mu\nu} \phi^{\alpha}. \Tag{(86.1)} \] Let \[ \Star{\{\mu\nu, \alpha\}} \equiv \{\mu\nu, \alpha\} - g_{\mu}^{\alpha} \kappa_{\nu} - g_{\nu}^{\alpha} \kappa_{\mu} + g_{\mu\nu} \kappa^{\alpha}. \Tag{(86.2)} \] Then by \Eq{(86.1)} and~\Eq{(85.52)} \[ \Star{\{\mu\nu, \alpha\}}' = \Star{\{\mu\nu, \alpha\}}. \Tag{(86.3)} \] The ``generalised $3$-index symbol'' $\Star{\{\mu\nu, \alpha\}}$ has the ``in-'' property, being unaltered by any gauge-transformation. It is, of course, not a tensor. We shall generally indicate by a star~(${}^{*}$) quantities generalised from corresponding expressions in Riemannian geometry in order to be independent of (or covariant with) the gauge-system. The following illustrates the general method of procedure.