Alternatively we can express a physical quantity of the second kind as ``so much per unit natural volume ($\sqrt{-g}\, d\tau$)''; it is then represented by a \PageSep{112} tensor. From the physical point of view it is perhaps as rational to express it in this way, as to express it by a tensor-density ``so much per unit mesh~($d\tau$).'' But analytically this is a somewhat hybrid procedure, because we seem to be employing simultaneously two systems of coordinates, the one openly for measuring the physical quantity, the other (a natural system) implicitly for measuring the volume containing it. It cannot be considered wrong in a physical sense to represent quantities of the second kind by tensors; but the analysis exposes our sub-conscious reference to~$\sqrt{-g}\, d\tau$, by the repeated appearance of~$\sqrt{-g}$ in the formulae. In any kind of space-time it is possible to choose coordinates such that $\sqrt{-g} = 1$ everywhere; for if three of the systems of partitions have been drawn arbitrarily, the fourth can be drawn so as to intercept meshes all of equal natural volume. In such coordinates tensors and tensor-densities become equivalent, and the algebra may be simplified; but although this simplification does not involve any loss of generality, it is liable to obscure the deeper significance of the theory, and it is not usually desirable to adopt it. \Section{50.}{The problem of the rotating disc} \index{Problem!of rotating disc}% \index{Rotating disc}% We may consider at this point a problem of some historic interest--- A disc made of homogeneous incompressible material is caused to rotate with angular velocity~$\omega$; to find the alteration in length of the radius. The old paradox associated with this problem---that the circumference moving longitudinally might be expected to contract, whilst the radius moving transversely is unaltered---no longer troubles us\footnotemark.\footnotetext {\Title{Space, Time and Gravitation}, p.~75.} But the general theory of relativity gives a quantitative answer to the problem, which was first obtained by Lorentz by a method different from that given here\footnotemark.\footnotetext {\Title{Nature}, vol.~106, p.~795.} We must first have a clear understanding of what is meant by the word \index{Incompressibility}% incompressible. Let us isolate an element of the rotating disc, and refer it to axes with respect to which it has no velocity or acceleration (proper-measure); then except for the fact that it is under stress due to the cohesive forces of surrounding matter, it is relatively in the same state as an element of the non-rotating disc referred to fixed axes. Now the meaning of \emph{incompressible} is that no stress-system can make any difference in the closeness of packing of the molecules; hence the particle-density a (referred to proper-measure) is the same as for an element of the non-rotating disc. But the particle-density~$\sigma'$ referred to axes fixed in space may be different.