In the transformation~\Eq{(15.3)} we have paid no attention to any contraction of the standards of length or retardation of clocks due to motion with the rotating axes. The formulae of transformation are those of elementary kinematics, so that $x_{1}'$, $x_{2}'$, $x_{3}'$, $x_{4}'$ are quite strictly the coordinates used in the ordinary theory of rotating axes. But it may be suggested that elementary kinematics is now seen to be rather crude, and that it would be worth while to touch up the formulae~\Eq{(15.3)} so as to take account of these small changes of the standards. A little consideration shows that the suggestion is impracticable. \PageSep{36} It was shown in \SecRef{4} that if $x_{1}'$, $x_{2}'$, $x_{3}'$, $x_{4}'$ represent rectangular coordinates and time as partitioned by direct readings of scales and clocks, then \[ ds^{2} = -dx_{1}'^{2} - dx_{2}'^{2} - dx_{3}'^{2} + c^{2}\, dx_{4}'^{2}, \Tag{(15.45)} \] so that coordinates which give any other formula for the interval cannot represent the immediate indications of scales and clocks. As shown at the end of \SecRef{5}, the only transformations which give \Eq{(15.45)} are Lorentz transformations. If we wish to make a transformation of a more general kind, such as that of~\Eq{(15.3)}, we must necessarily abandon the association of the coordinate-system with uncorrected scale and clock readings. It is useless to try to ``improve'' the transformation to rotating axes, because the supposed improvement could only lead us back to a coordinate-system similar to the fixed axes with which we started. The inappropriateness of rotating axes to scale and clock measurements can be regarded from a physical point of view. We cannot keep a scale or clock at rest in the rotating system unless we constrain it, i.e.\ subject it to molecular bombardment---an ``outside influence'' whose effect on the measurements must not be ignored. In the $x$, $y$, $z$, $t$ system of coordinates the scale and clock are the natural equipment for exploration. In other systems they will, if unconstrained, continue to measure~$ds$; but the reading of~$ds$ is no longer related in a simple way to the differences of coordinates which we wish to determine; it depends on the more complicated calculations involved in~\Eq{(2.1)}. The scale and clock to some extent lose their pre-eminence, and since they are rather elaborate appliances it may be better to refer to some simpler means of exploration. We consider then two simpler test-objects---the moving particle and the \index{Particle!motion of}% \index{Track of moving particle and light-pulse}% light-pulse.