In ordinary rectangular coordinates and time $x$,~$y$,~$z$, $t$ an undisturbed particle moves with uniform velocity, so that its track is given by the equations \[ x = a + bt,\quad y = c + dt,\quad z = e + ft, \Tag{(15.5)} \] i.e.\ the equations of a straight line in four dimensions. By substituting from~\Eq{(15.3)} we could find the equations of the track in rotating coordinates; or by substituting from~\Eq{(15.2)} we could obtain the differential equations for any desired coordinates. But there is another way of proceeding. The differential equations of the track may be written \[ \frac{d^{2} x}{ds^{2}},\quad \frac{d^{2} y}{ds^{2}},\quad \frac{d^{2} z}{ds^{2}},\quad \frac{d^{2} t}{ds^{2}} = 0, \Tag{(15.6)} \] which on integration, having regard to the condition~\Eq{(7.1)}, give equations~\Eq{(15.5)}. The equations~\Eq{(15.6)} are comprised in the single statement \[ \int ds \text{ is stationary} \Tag{(15.7)} \] for all arbitrary small variations of the track which vanish at the initial and final limits---a well-known property of the straight line. \PageSep{37} In arriving at~\Eq{(15.7)} we use freely the geometry of the $x$, $y$, $z$, $t$ system \index{Abstract geometry and natural geometry}% \index{Geometry, Riemannian!abstract and natural}% given by~\Eq{(7.1)}; but the final result does not allude to coordinates at all, and must be unaltered whatever system of coordinates we are using. To obtain explicit equations for the track in any desired system of coordinates, we substitute in~\Eq{(15.7)} the appropriate expression~\Eq{(2.1)} for~$ds$ and apply the calculus of variations. The actual analysis will be given in \SecRef{28}. The track of a light-pulse, being a straight line in four dimensions, will \index{Light-pulse!equation of track}% also satisfy~\Eq{(15.7)}; but the light-pulse has the special velocity~$c$ which gives the additional condition obtained in \SecRef{7}, viz\Add{.}\ \[ ds = 0. \Tag{(15.8)} \] Here again there is no reference to any coordinates in the final result. We have thus obtained equations \Eq{(15.7)} and \Eq{(15.8)} for the behaviour of the moving particle and light-pulse which must hold good whatever the coordinate-system chosen. The indications of our two new test-bodies are connected with the interval, just as in \SecRef{3} the indications of the scale and clock were connected with the interval. It should be noticed however that whereas the use of the older test-bodies depends only on the truth of the fundamental axiom, the use of the new test-bodies depends on the truth of the empirical laws of motion and of light-propagation. In a deductive theory this appeal to empirical laws is a blemish which we must seek to remove later. \Section{16.}{Fields of force} \index{Fields of force}%