To sum up the results of this section, if we choose coordinates such that the general quadratic form reduces to \[ ds^{2} = dy_{1}^{2} + dy_{2}^{2} + dy_{3}^{2} + dy_{4}^{2}, \Tag{(4.95)} \] then $y_{1}$, $y_{2}$, $y_{3}$ and $y_{4} \sqrt{-1}$ will represent ordinary rectangular coordinates and time. If we choose coordinates for which \[ ds^{2} = dy_{1}^{2} + dy_{2}^{2} + dy_{3}^{2} + dy_{4}^{2} + 2\alpha\, dy_{1}\, dy_{4} + 2\beta\, dy_{2}\, dy_{4} + 2\gamma\, dy_{3}\, dy_{4}, \Tag{(4.96)} \] these coordinates also will agree with rectangular coordinates and time so far as the more primitive notions of time are concerned; but the reckoning by this formula of differences of time at different places will not agree with the reckoning adopted in physics and astronomy according to long established practice. For this reason it would only introduce confusion to admit these coordinates as a permissible space and time system. We who regard all coordinate-frames as equally fictitious structures have no special interest in ruling out the more general form~\Eq{(4.96)}. It is not a question of ascribing greater significance to one frame than to another, but of discovering which frame corresponds to the space and time reckoning generally accepted and used in standard works such as the Nautical Almanac. As far as \SecRef{14} our work will be subject to the condition that we are dealing with a region of the world in which the $g$'s are constant, or approximately constant. A region having this property is called \emph{flat}. The theory of this \index{Flat space-time}% \index{Special theory of relativity}% case is called the ``special'' theory of relativity; it was discussed by Einstein in 1905---some ten years before the general theory. But it becomes much simpler when regarded as a special case of the general theory, because it is no longer necessary to defend the conditions for its validity as being essential properties of space-time. For a given region these conditions may hold, or they may not. The special theory applies only if they hold; other cases must be referred to the general theory. \PageSep{17} \Section{5.}{The Lorentz transformation} \index{Lorentz transformation}% Make the following transformation of coordinates \index{Transformation of coordinates!Lorentz}% \begin{gather*} x = \beta(x' - ut')),\ y = y',\ z = z',\ t = \beta (t' - ux'/c^{2}), \Tag{(5.1)} \\ \beta = (1 - u^{2}/c^{2})^{-\frac{1}{2}}, \end{gather*} where $u$~is any real constant not greater than~$c$. We have by~\Eq{(5.1)} \begin{align*} dx^{2} - c^{2}\, dt^{2} &= \beta^{2} \bigl\{(dx' - u\, dt')^{2} - c^{2} (dt' - u\, dx'/c^{2})^{2}\bigr\}\displaybreak[0] \\ &= \beta^{2} \left\{\left(l - \frac{u^{2}}{c^{2}}\right) dx'^{2} - (c^{2} - u^{2})\, dt'^{2}\right\} \\ &= dx'^{2} - c^{2}\, dt'^{2}. \end{align*} Hence from \Eq{(4.6)} \[ ds^{2} = dx^{2} + dy^{2} + dz^{2} - c^{2}\, dt^{2} = dx'^{2} + dy'^{2} + dz'^{2} - c^{2}\, dt'^{2}. \Tag{(5.2)} \]