If %[** TN: First group not broken in the original] \begin{gather*} x_{1} = f_{1}(x_{1}', x_{2}', x_{3}', x_{4}');\quad x_{2} = f_{2}(x_{1}', x_{2}', x_{3}', x_{4}'); \text{ etc.,} \\ dx_{1} = \frac{\dd f_{1}}{\dd x_{1}'}\, dx_{1}' + \frac{\dd f_{1}}{\dd x_{2}'}\, dx_{2}' + \frac{\dd f_{1}}{\dd x_{3}'}\, dx_{3}' + \frac{\dd f_{1}}{\dd x_{4}'}\, dx_{4}'; \text{ etc.,} \Tag{(15.1)} \end{gather*} or it may be written simply, \[ dx_{1} = \frac{\dd x_{1}}{\dd x_{1}'}\, dx_{1}' + \frac{\dd x_{1}}{\dd x_{2}'}\, dx_{2}' + \frac{\dd x_{1}}{\dd x_{3}'}\, dx_{3}' + \frac{\dd x_{1}}{\dd x_{4}'}\, dx_{4}'; \text{ etc.,} \Tag{(15.2)} \] Substituting from \Eq{(15.2)} in~\Eq{(2.1)} we see that $ds^{2}$~will be a homogeneous quadratic function of the differentials of the new coordinates; and the new coefficients $g_{11}'$, $g_{22}'$, etc.\ could be written down in terms of the old, if desired. For an example consider the usual transformation to axes revolving with \index{Rotating axes, quadratic form for}% constant angular velocity~$\omega$, viz. \[ \left. \begin{aligned} x &= x_{1}' \cos \omega x_{4}' - x_{2}' \sin \omega x_{4}' \\ y &= x_{1}' \sin \omega x_{4}' + x_{2}' \cos \omega x_{4}' \\ z &= x_{3}' \\ t &= x_{4}' \end{aligned} \right\}. \Tag{(15.3)} \] Hence \begin{align*} dx &= dx_{1}' \cos \omega x_{4}' - dx_{2}' \sin \omega x_{4}' + \omega(-x_{1}' \sin \omega x_{4}' - x_{2}' \cos \omega x_{4}')\, dx_{4}', \\ dy &= dx_{1}' \sin \omega x_{4}' + dx_{2}' \cos \omega x_{4}' + \omega( x_{1}' \cos \omega x_{4}' - x_{2}' \sin \omega x_{4}')\, dx_{4}', \\ dz &= dx_{3}', \\ dt &= dx_{4}'. \end{align*} Taking units of space and time so that $c = 1$, we have for our original fixed coordinates by~\Eq{(7.1)} \[ ds^{2} = -dx^{2} - dy^{2} - dz^{2} + dt^{2}. \] Hence, substituting the values found above, \begin{multline*} ds^{2} = - dx_{1}'^{2} - dx_{2}'^{2} - dx_{3}'^{2} + \bigl\{1 - \omega^{2} (x_{1}'^{2} + x_{2}'^{2})\bigr\}\, dx_{4}'^{2} \\ + 2\omega x_{2}'\, dx_{1}'\, dx_{4}' - 2\omega x_{1}'\, dx_{2}'\, dx_{4}'. \Tag{(15.4)} \end{multline*} Remembering that all observational differences of co\-ord\-inate-systems must arise \Foreign{via} the interval, this formula must comprise everything which distinguishes the rotating system from a fixed system of coordinates.