translation. More interestingly, from $\delta_{\alpha}(p) = d_1$ and $\cos(\alpha,\beta)=\cos\theta$, one can partially solve the equations for the components of the position of $C_0$. The resulting expressions are shown in Table \[Tb:Positions\]. Observe that for each class we have two solutions in terms of the free parameter $|a|$. For the classes ${\mathcal{G}_3}-{\mathcal{G}_5}$ the two solutions are those for which $d_1 \!\cos\theta$ is given by the fourth column in Table \[Tb:Express\].

Two remarks are in order here. First, it is convenient to write down the components of the position of $C_0$ in terms of the parameter $|a|$, because this parameter can be easily determined once two more triples of multiple images, say $\{D_0,D_1,D_2\}$ and $\{E_0,E_1,E_2\}$, in the neighbourhood of $\{C_0,C_1,C_2\}$ have been identified, as shown in Section \[Decide\].[^4] Once this has been done, the positions of $C_0$, $D_0$ and $E_0$ can be used to predict multiple images of them due to the inverse isometry $\alpha^{-1}$, thus yielding a definitive observational test for the hypothesis of the multiply connectedness of our Universe. Second, note that the $x-\,$coordinate is not constrained by this configuration of topological images. This freedom of the $x-\,$coordinate is a consequence of homogeneity of manifolds of classes ${\mathcal{G}_2}-{\mathcal{G}_5}$ along the 