We have to keep side by side the two methods of describing the configurations of events by coordinates and by the mutual intervals, respectively---the first for its conciseness, and the second for its immediate absolute significance. It is therefore necessary to connect the two modes of description by a formula which will enable us to pass readily from one to the other. The particular formula will depend on the coordinates chosen as well as on the absolute properties of the region of the world considered; but it appears that in all cases the formula is included in the following general form--- The interval $ds$ between two neighbouring events with coordinates \index{Quadratic formula for interval}% $(x_{1}, x_{2}, x_{3}, x_{4})$ and $(x_{1} + dx_{1}, x_{2} + dx_{2}, x_{3} + dx_{3}, x_{4} + dx_{4})$ in any coordinate-system is given by %[** TN: Reformatted from the original] \begin{multline*} ds^{2} = g_{11}\, dx_{1}^{2} + g_{22}\, dx_{2}^{2} + g_{33}\, dx_{3}^{2} + g_{44}\, dx_{4}^{2} \\ \begin{aligned}[b] &+ 2g_{12}\, dx_{1}\, dx_{2} + 2g_{13}\, dx_{1}\, dx_{3} + 2g_{14}\, dx_{1}\, dx_{4} \\ &+ 2g_{23}\, dx_{2}\, dx_{3} + 2g_{24}\, dx_{2}\, dx_{4} + 2g_{34}\, dx_{3}\, dx_{4}, \end{aligned} \Tag{(2.1)} \end{multline*} where the coefficients $g_{11}$, etc. are functions of $x_{1}$,~$x_{2}$, $x_{3}$,~$x_{4}$. That is to say, $ds^{2}$~is some quadratic function of the differences of coordinates.% \PageSep{11} This is, of course, not the most general case conceivable; for example, we might have a world in which the interval depended on a general quartic function of the~$dx$'s. But, as we shall presently see, the quadratic form~\Eq{(2.1)} is definitely indicated by observation as applying to the actual world. Moreover near the end of our task (\SecRef{97}) we shall find in the general theory of relation-structure a precise reason why a quadratic function of the coordinate-differences should have this paramount importance. Whilst the form of the right-hand side of~\Eq{(2.1)} is that required by observation, the insertion of~$ds^{2}$ on the left, rather than some other function of~$ds$, is merely a convention. The quantity~$ds$ is a measure of the interval. \index{Length!measurement of}% \index{Measure of interval}% It is necessary to consider carefully how measure-numbers are to be affixed to the different intervals occurring in nature. We have seen in the last section that equality of intervals can be tested observationally; but so far as we have yet gone, intervals are merely either equal or unequal, and their differences have not been further particularised. Just as wind-strength may be measured by velocity, or by pressure, or by a number on the Beaufort scale, so the relation of extension between two events could be expressed numerically according to many different plans. To conform to~\Eq{(2.1)} a particular code of measure-numbers must be adopted; the nature and advantages of this code will be explained in the next section.