We might write down at once by~\Eq{(14.1)} \[ \sigma' = \sigma (1 - \omega^{2} r^{2})^{-\frac{1}{2}}, \] since $\omega r$~is the velocity of the element. This would in fact give the right result. But in \SecRef{14} acceleration was not taken into account and we ought to \PageSep{113} proceed more rigorously. We use the accented coordinates of \SecRef{15} for our rotating system, and easily calculate from~\Eq{(15.4)} that \[ \sqrt{-g'} = 1, \] and since $x_{1}'$, $x_{2}'$, $x_{3}'$ are constant for an element of the disc, the proper-time \[ ds = \Chg{\surd\bigl(1 - \omega^{2}(x_{1}'^{2} + x_{2}'^{2})\bigr)\, dx_{4}'} {\sqrt{1 - \omega^{2}(x_{1}'^{2} + x_{2}'^{2})}\, dx_{4}'}. \] If $dW$~is the proper-volume of the element, by~\Eq{(49.42)} \[ dW\, ds = \sqrt{-g'} · dx_{1}'\, dx_{2}'\, dx_{3}'\, dx_{4}'. \] Hence \begin{align*} dW &= \bigl(1 - \omega^{2}(x_{1}'^{2} + x_{2}'^{2})\bigr)^{-\frac{1}{2}}\, dx_{1}'\, dx_{2}'\, dx_{3}' \\ &= (1 - \omega^{2} r^{2})^{-\frac{1}{2}}\, r'\, dr'\, d\theta'\, dx_{3}'. \end{align*} If the thickness of the disc is $\delta x_{3}' = b$, and its boundary is given by $r' = a'$, the total number of particles in the disc will be \[ N = \int \sigma\, dW = 2\pi\sigma b \int_{0}^{a'} (1 - \omega^{2} r'^{2})^{-\frac{1}{2}}\, r'\, dr'. \] Since this number is unaltered by the rotation, $a'$~must be a function of~$\omega$ such that \[ \int_{0}^{a'} (1 - \omega^{2} r^{2})^{-\frac{1}{2}}\, r'\, dr' = \text{const.}, \] or \[ \Chg{\frac{1}{\omega^{2}}\bigl(1 - \surd(1 - \omega^{2} a'^{2})\bigr)} {\frac{1}{\omega^{2}}\bigl(1 - \sqrt{1 - \omega^{2} a'^{2}}\bigr)} = \text{const.} \] Expanding the square-root, this gives approximately \[ \tfrac{1}{2} a'^{2} (1 + \tfrac{1}{4} \omega^{2} a'^{2}) = \text{const.}, \] so that if $a$~is the radius of the disc at rest \[ a' = (1 + \tfrac{1}{8} \omega^{2} a'^{2}) = a. \] Hence to the same approximation \[ a' = a (1 - \tfrac{1}{8} \omega^{2} a^{2}). \] Note that $a'$~is the radius of the rotating disc according to measurement with fixed scales, since the rotating and non-rotating coordinates have been connected by the elementary transformation~\Eq{(15.3)}. We see that the contraction is one quarter of that predicted by a crude application of the FitzGerald formula to the circumference. \Section{51.}{The divergence of a tensor} \index{Contracted derivative (divergence)}% \index{Divergence!tensor@of a tensor}% In the elementary theory of vectors the divergence \[ \frac{\dd X}{\dd x} + \frac{\dd Y}{\dd y} + \frac{\dd Z}{\dd z} \] is important; we can to some extent grasp its geometrical significance. In our general notation, this expression becomes \[ \frac{\dd A^{\mu}}{\dd x_{\mu}}. \] \PageSep{114} But evidently a more fundamental operation is to take the covariant derivatives which will give an invariant \[ (A^{\mu})_{\mu}. \] We therefore define the \emph{divergence} of a tensor as its contracted covariant derivative.