The accented and unaccented coordinates give the same formula for the interval, so that the intervals between corresponding pairs of mesh-corners will be equal, and therefore in all observable respects they will be alike. We shall recognise $x'$,~$y'$,~$z'$ as rectangular coordinates in space, and $t'$~as the associated time. We have thus arrived at another possible way of reckoning space and time---another fictitious space-time frame, equivalent in all its properties to the original one. For convenience we say that the first reckoning is that of an observer~$S$ and the second that of an observer~$S'$, both observers being at rest in their respective spaces\footnotemark.\footnotetext {This is partly a matter of nomenclature. A sentient observer can force himself to ``recollect that he is moving'' and so adopt a space in which he is not at rest; but he does not so readily adopt the time which properly corresponds; unless he uses the space-time frame in which he is at rest, he is likely to adopt a hybrid space-time which leads to inconsistencies. There is no ambiguity if the ``observer'' is regarded as merely an involuntary measuring apparatus, which by the principles of \SecRef{4} naturally partitions a space and time with respect to which it is at rest.} The constant~$u$ is easily interpreted. Since $S$~is at rest in his own space, his location is given by $x = \text{const}$. By~\Eq{(5.1)} this becomes, in $S'$'s~coordinates, $x' - ut' = \text{const.}$; that is to say, $S$~is travelling in the $x'$-direction with velocity~$u$. Accordingly the constant~$u$ is interpreted as the velocity of~$S$ relative to~$S'$. It does not follow immediately that the velocity of~$S'$ relative to~$S$ is~$-u$; but this can be proved by algebraical solution of the equations~\Eq{(5.1)} to determine $x'$,~$y'$, $z'$,~$t'$. We find \[ x' = \beta(x + ut),\quad y' = y,\quad z' = z,\quad t' = \beta(t + ux/c^{2}), \Tag{(5.3)} \] showing that an interchange of $S$ and~$S'$ merely reverses the sign of~$u$.