An \emph{isolated vector~$A_{\mu}$} may be taken at the starting point and carried by parallel displacement round the circuit, leading to a determinate value of~$\delta A_{\mu}$. \PageSep{71} \index{Geodesic, equations of!produced by parallel displacement}% \index{Velocity-vector}% In~\Eq{(33.3)} this is expressed in terms of derivatives of a \emph{vector-field~$A_{\mu}$} extending throughout the region of integration. For a large circuit this would involve values of~$A_{\mu}$ remote from the initial vector, which are obviously irrelevant to the calculation of~$\delta A_{\mu}$. It is rather remarkable that there should exist such a formula even for an infinitesimal circuit; the fact is that although $A_{\mu\nu\sigma} - A_{\mu\sigma\nu}$ at a point formally refers to a vector-field, its value turns out to depend solely on the isolated vector~$A_{\mu}$ (see equation~\Eq{(34.3)}). The contravariant vector~$dx_{\mu}/ds$ gives a direction in the four-dimensional world which is interpreted as a velocity from the ordinary point of view which separates space and time. We shall usually call it a ``velocity''; its connection with the usual three-dimensional vector $(u, v, w)$ is given by \[ \frac{dx_{\mu}}{ds} = \beta(u, v, w, 1), \] where $\beta$~is the FitzGerald factor~$dt/ds$. The length~\Eq{(26.5)} of a velocity is always unity. If we transfer $dx_{\mu}/ds$ continually along itself by parallel displacement we obtain a geodesic. For by~\Eq{(29.4)} the condition for parallel displacement is \[ \frac{\dd}{\dd x_{\nu}} \left(\frac{\dd x_{\mu}}{ds}\right) + \{\alpha\nu, \mu\}\, \frac{\dd x_{\alpha}}{ds} = 0. \] Hence multiplying by~$dx_{\nu}/ds$ \[ \frac{\dd^{2} x_{\mu}}{ds^{2}} + \{\alpha\nu, \mu\}\, \frac{\dd x_{\alpha}}{ds}\, \frac{dx_{\nu}}{ds} = 0, \Tag{(33.4)} \] which is the condition for a geodesic~\Eq{(28.5)}. Thus in the language used at the beginning of this section, a geodesic is a line in four dimensions whose direction undergoes no absolute change. \Section{34.}{The Riemann-Christoffel tensor}