Let $A_{\mu}^{\nu}$~be a symmetrical in-tensor; its divergence~\Eq{(51.31)} becomes on gauge-transformation \begin{align*} {A'}_{\mu\nu}^{\nu} &= \frac{1}{\lambda^{4} \sqrt{-g}}\, \frac{\dd}{\dd x_{\nu}} (A_{\mu}^{\nu} \lambda^{4} \sqrt{-g}) - \tfrac{1}{2} (\lambda^{-2} A^{\alpha\beta})\, \frac{\dd}{\dd x_{\mu}} (\lambda^{2} g_{\alpha\beta}) \\ &= \frac{1}{\sqrt{-g}}\, \frac{\dd}{\dd x_{\nu}} (A_{\mu}^{\nu}\sqrt{-g}) %[** TN: Not broken in the original] \begin{aligned}[t] &- \tfrac{1}{2} A^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{\mu}} + A_{\mu}^{\nu} · \frac{1}{\lambda^{4}}\, \frac{\dd\lambda^{4}}{\dd x_{\nu}} \\ &- \tfrac{1}{2} A^{\alpha\beta} g_{\alpha\beta} · \frac{1}{\lambda^{2}}\, \frac{\dd\lambda^{2}}{\dd x_{\mu}} \end{aligned} \\ &= A_{\mu\nu}^{\nu} + 4A_{\mu}^{\nu} \phi_{\nu} - A\phi_{\mu}. \end{align*} Hence by~\Eq{(85.52)} the quantity \[ \Star{A}_{\mu\nu}^{\nu} = A_{\mu\nu}^{\nu} - 4A_{\mu}^{\nu} \kappa_{\nu} + A\kappa_{\mu} \Tag{(86.4)} \] is unaltered by any gauge-transformation, and is accordingly an in-vector. This operation may be called in-covariant differentiation, and the result is the in-divergence. The result is modified if $A^{\mu\nu}$~is the in-tensor, so that $A_{\mu}^{\nu}$~is a co-tensor. The different associated tensors are not equally fundamental in Weyl's geometry, since only one of them can be an in-tensor. Unless expressly stated a final suffix will indicate ordinary covariant (not in-covariant) differentiation. \PageSep{204} \Section{87.}{The generalised Riemann-Christoffel tensor} \index{Riemann-Christoffel tensor!generalisation of}% Corresponding to~\Eq{(34.4)} we write \[ \Star{B}_{\mu\nu\sigma}^{\epsilon} \!=\! -\frac{\dd}{\dd x_{\sigma}} \Star{\{\mu\nu, \epsilon\}} + \Star{\{\mu\sigma, \alpha\}} \Star{\{\alpha\nu, \epsilon\}} + \frac{\dd}{\dd x_{\nu}} \Star{\{\mu\sigma, \epsilon\}} - \Star{\{\mu\nu, \alpha\}} \Star{\{\alpha\sigma, \epsilon\}}. \Tag{(87.1)} \] This will be an in-tensor since the starred symbols are all independent of the gauge; and it will be evident when we reach~\Eq{(87.4)} that the generalisation has not destroyed the ordinary tensor properties.