We endeavoured to explain how this change of direction can occur in a curved world by the example of a ship sailing on a curved ocean (\SecRef{33}). Having convinced ourselves that there is no logical impossibility in the result that the direction changes, we cannot very well see anything self-contradictory in the length changing also. It is true that we have just given a mathematical proof that the length does not change; but that only means that a change of length is excluded by conditions which have been introduced, perhaps inadvertently, in the postulates of Riemannian geometry. We can construct a geometry in which the change of length occurs, without landing ourselves in a contradiction. In the more general geometry, we have in place of~\Eq{(84.1)} \[ \delta A_{\mu} = \tfrac{1}{2}\, \Star{B}_{\mu\nu\sigma\epsilon} A^{\epsilon}\, dS^{\nu\sigma}, \Tag{(84.21)} \] \PageSep{199} where $\Star{B}_{\mu\nu\sigma\epsilon}$~is a more general tensor which is \emph{not} antisymmetrical in $\mu$ and~$\epsilon$. It will be antisymmetrical in $\nu$ and~$\sigma$ since a symmetrical part would be meaningless in~\Eq{(84.21)}, and disappear owing to the antisymmetry of~$dS^{\nu\sigma}$. Writing \begin{gather*} R_{\mu\nu\sigma\epsilon} = \tfrac{1}{2}(\Star{B}_{\mu\nu\sigma\epsilon} - \Star{B}_{\epsilon\nu\sigma\mu});\quad F_{\mu\nu\sigma\epsilon} = \tfrac{1}{2}(\Star{B}_{\mu\nu\sigma\epsilon} + \Star{B}_{\epsilon\nu\sigma\mu}), \\ \delta A_{\mu} = \tfrac{1}{2}(R_{\mu\nu\sigma\epsilon} + F_{\mu\nu\sigma\epsilon}) A^{\epsilon}\, dS^{\nu\sigma}, \Tag{(84.22)} \end{gather*} where $R$~is antisymmetrical, and $F$~symmetrical, in $\mu$ and~$\epsilon$. Then the change of length~$l$ is given by \[ \delta(l^{2}) = 2A^{\mu}\, \delta A_{\mu} = F_{\mu\nu\sigma\epsilon} A^{\mu} A^{\epsilon}\, dS^{\nu\sigma}, \Tag{(84.3)} \] which does not vanish. To obtain Weyl's geometry we must impose two restrictions on~$F_{\mu\nu\sigma\epsilon}$: \\ \Indent\Item{(a)} $F_{\mu\nu\sigma\epsilon}$~is of the special form~$g_{\mu\epsilon}F_{\nu\sigma}$, \\ \Indent\Item{(b)} $F_{\nu\sigma}$~is the curl of a vector. The second restriction is logically necessary. We have expressed the change of a vector taken round a circuit by a formula involving a surface bounded by the circuit. We may choose different surfaces, all bounded by the same circuit; and these have to give the same result for~$\delta A_{\mu}$. It is easily seen, as in Stokes's theorem, that these results will only be consistent if the co-factor of~$dS^{\nu\sigma}$ is a curl.