The essential property of the foregoing transformation is that it leaves the formula for~$ds^{2}$ unaltered~\Eq{(5.2)}, so that the coordinate-systems which it connects are alike in their properties. Looking at the matter more generally, we have already noted that the reduction to the sum of four squares can be made in many ways, so that we can have \[ ds^{2} = dy_{1}^{2} + dy_{2}^{2} + dy_{3}^{2} + dy_{4}^{2} = dy_{1}'^{2} + dy_{2}'^{2} + dy_{3}'^{2} + dy_{4}'^{2}. \Tag{(5.4)} \] \PageSep{18} The determination of the necessary connection between any two sets of coordinates satisfying this equation is a problem of pure mathematics; we can use freely the conceptions of four-dimensional geometry and imaginary rotations to find this connection, whether the conceptions have any physical significance or not. We see from~\Eq{(5.4)} that $ds$~is the distance between two points in four-dimensional Euclidean space, the coordinates $(y_{1}, y_{2}, y_{3}, y_{4})$ and $(y_{1}', y_{2}', y_{3}', y_{4}')$ being rectangular systems (real or imaginary) in that space. Accordingly these coordinates are related by the general transformations from one set of rectangular axes to another in four dimensions, viz.\ translations and rotations. Translation, or change of origin, need not detain us; nor need a rotation of the space-axes $(y_{1}, y_{2}, y_{3})$ leaving time unaffected. The interesting case is a rotation in which $y_{4}$~is involved, typified by \[ y_{1} = y_{1}' \cos\theta - y_{4}' \sin\theta,\quad y_{4} = y_{1}' \sin\theta + y_{4}' \cos\theta. \] Writing $u = ic \tan\theta$, so that $\beta = \cos\theta$, this leads to the Lorentz transformation~\Eq{(5.1)}. Thus, apart from obvious trivial changes of axes, the Lorentz transformations are the only ones which leave the form~\Eq{(4.6)} unaltered. Historically this transformation was first obtained for the particular case of electromagnetic equations. Its more general character was pointed out by Einstein in 1905. \Section{6.}{The velocity of light} \index{Addition of velocities}% Consider a point moving along the $x$-axis whose velocity measured by~$S'$ is~$v'$, so that \[ v' = \frac{dx'}{dt'}. \Tag{(6.1)} \] Then by \Eq{(5.1)} its velocity measured by~$S$ is \begin{align*} v = \frac{dx}{dt} &= \frac{\beta(dx' - u\, dt')}{\beta(dt' - u\, dx'/c^{2})} \\ &= \frac{v' - u}{1 - uv'/c^{2}}\quad\text{by \Eq{(6.1)}.} \Tag{(6.2)} \end{align*} In non-relativity kinematics we should have taken it as axiomatic that $v = v' - u$. If two points move relatively to~$S'$ with equal velocities in opposite directions $+v'$ and~$-v'$, their velocities relative to~$S$ are \[ \frac{v' - u}{1 - uv'/c^{2}}\quad\text{and}\quad -\frac{v' + u}{1 + uv'/c^{2}}. \] As we should expect, these speeds are usually unequal; but there is an exceptional case when $v' = c$. The speeds relative to~$S$ are then also equal, both in fact being equal to~$c$. \PageSep{19}