Suppose that in this volume there is only a single particle, so that the energy-tensor vanishes everywhere except in a narrow tube. By~\Eq{(56.1)} the quadruple integral becomes \[ -\iiiint \{\alpha\nu, \mu\} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\nu}}{ds}\, \rho_{0} \sqrt{-g}\, d\tau = -\{\alpha\beta, \mu\} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}\, m\, ds, \Tag{(56.4)} \] since $\rho_{0} \sqrt{-g}\, d\tau = \rho_{0}\, dW · ds = dm · ds$, where $dm$~is the proper-mass. On the left the triple integrals vanish except at the two points where the world-line intersects the boundary of the region. For convenience we draw the boundary near these two points in the planes $dx_{1} = 0$, so that only the first of the four integrals survives. The left-hand side of~\Eq{(56.3)} becomes \[ \left[\iiint \rho_{0} \sqrt{-g}\, \frac{dx_{\mu}}{ds}\, \frac{dx_{1}}{ds}\, dx_{2}\, dx_{3}\, dx_{4}\right], \Tag{(56.51)} \] the bracket denoting the difference at the two ends of the world-line. The geometrical volume of the oblique cylinder cut off from the tube by sections $dx_{2}\, dx_{3}\, dx_{4}$ at a distance apart~$ds$ measured along the tube is \[ \frac{dx_{1}}{ds} · ds\, dx_{2}\, dx_{3}\, dx_{4}. \] %\PageSep{127} Multiplying by $\rho_{0} \sqrt{-g}$ we get the amount of~$\rho_{0}$ contained\footnotemark,\footnotetext {The amount of density in a four-dimensional volume is, of course, not the mass but a quantity of dimensions $\text{mass} \times \text{time}$.} which is~$dm\, ds$. Hence \Eq{(56.51)}~reduces to \[ \left[m\, \frac{dx_{\mu}}{ds}\right]. \] The difference at the two limits is \[ \frac{d}{ds} \left(m\, \frac{dx_{\mu}}{ds}\right) ds, \Tag{(56.52)} \] where $ds$~is now the length of track between the two limits as in~\Eq{(56.4)}. By \Eq{(56.4)} and~\Eq{(56.52)} the equation reduces to \[ \frac{d}{ds} \left(m\, \frac{dx_{\mu}}{ds}\right) = -m \{\alpha\beta, \mu\}\, \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}. \Tag{(56.6)} \] Provided that $m$~is constant this gives the equations of a geodesic \Eq{(28.5)}, showing that the track of an isolated particle is a geodesic. The constancy of~$m$ can be proved formally as follows---