\emph{The only fundamental tensors which do not contain derivatives of~$g_{\mu\nu}$ beyond the second order are functions of $g_{\mu\nu}$ and~$B_{\mu\nu\sigma}^{\epsilon}$.} \PageSep{80} This shows that our treatment of the tensors describing the character of space-time has been exhaustive as far as the second order. If for suitably chosen coordinates two surfaces have the same $g_{\mu\nu}$ and $B_{\mu\nu\sigma}^{\epsilon}$ at some point, they will be applicable to one another as far as cubes of the coordinates; the two tensors suffice to specify the whole metric round the point to this extent. Having made the first derivatives vanish, we can by the linear transformation explained in \SecRef{4} give the $g_{\mu\nu}$ Galilean values at the selected point. The coordinates so obtained are called \emph{natural coordinates} at the point and \index{Natural coordinates}% \index{Natural coordinates!measure}% quantities referred to these coordinates are said to be expressed in \emph{natural measure}. Natural coordinates are thus equivalent to Galilean coordinates when only the $g_{\mu\nu}$ and their first derivatives are considered; the difference appears when we study phenomena involving the second derivatives. By making a Lorentz transformation (which leaves the coordinates still a natural system) we can reduce to rest the material located at the point, or an observer supposed to be stationed with his measuring appliances at the point. The \emph{natural measure} is then further particularised as the \emph{proper-measure} of the material, or observer. It may be noticed that the material will be at rest both as regards velocity and acceleration (unless it is acted on by electromagnetic forces) because there is no field of acceleration relative to natural coordinates. To sum up this discussion of special systems of coordinates.---When the Riemann-Christoffel tensor vanishes, we can adopt Galilean coordinates throughout the region. When it does not vanish we can adopt coordinates which agree with Galilean coordinates at a selected point in the values of the~$g_{\mu\nu}$ and their first derivatives but not in the second derivatives; these are called \emph{natural coordinates} at the point. Either Galilean or natural coordinates can be subjected to Lorentz transformations, so that we can select a system with respect to which a particular observer is at rest; this system will be the \emph{proper-coordinates} for that observer. Although we cannot in general make \index{Proper-coordinates}% natural coordinates agree with Galilean coordinates in the second derivatives of the~$g_{\mu\nu}$, we can impose $80$~partially arbitrary conditions on the $100$~second derivatives; and when these conditions are selected as in~\Eq{(36.7)} the resulting coordinates have been called \emph{canonical}.