Since the number of interchanges of the denominators is the same as the number of interchanges of the numerators \[ \frac{\epsilon_{\nu\xi\omicron\varpi}}{\epsilon_{\alpha\beta\gamma\delta}} = ±1 = \frac{\epsilon_{\iota\kappa\lambda\mu}}{\epsilon_{iklm}}, \Tag{(48.43)} \] so that the result of the transposition is \[ \epsilon_{\alpha\beta\gamma\delta}\, \epsilon_{\iota\kappa\lambda\mu}\, \frac{\dd x_{\nu}'}{\dd x_{\iota}}\, \frac{\dd x_{\xi}'}{\dd x_{\kappa}}\, \frac{\dd x_{\omicron}'}{\dd x_{\lambda}}\, \frac{\dd x_{\varpi}'}{\dd x_{\mu}} = \epsilon_{\nu\xi\omicron\varpi}\, \epsilon_{iklm}\, \frac{\dd x_{\alpha}'}{\dd x_{i}}\, \frac{\dd x_{\beta}'}{\dd x_{k}}\, \frac{\dd x_{\gamma}'}{\dd x_{l}}\, \frac{\dd x_{\delta}'}{\dd x_{m}}. \Tag{(48.5)} \] Making a similar transposition of the last four terms, \Eq{(48.41)}~becomes \begin{multline*} (4!)^{3} J^{2} g' = g_{\alpha\epsilon}'\, g_{\beta\zeta}'\, g_{\gamma\eta}'\, g_{\delta\theta}' · \frac{\dd x_{\alpha}'}{\dd x_{i}}\, \frac{\dd x_{\beta}'}{\dd x_{k}}\, \frac{\dd x_{\gamma}'}{\dd x_{l}}\, \frac{\dd x_{\delta}'}{\dd x_{m}} · \frac{\dd x_{\epsilon}'}{\dd x_{r}}\, \frac{\dd x_{\zeta}'}{\dd x_{s}}\, \frac{\dd x_{\eta}'}{\dd x_{t}}\, \frac{\dd x_{\theta}'}{\dd x_{u}} \\ · \epsilon_{iklm}\, \epsilon_{\nu\xi\omicron\varpi}\, \epsilon_{\nu\xi\omicron\varpi}\, \epsilon_{rstu}\, \epsilon_{\phi\chi\psi\omega}\, \epsilon_{\phi\chi\psi\omega}. \end{multline*} But by~\Eq{(23.22)} \[ g_{\alpha\epsilon}'\, \frac{\dd x_{\alpha}'}{\dd x_{i}}\, \frac{\dd x_{\epsilon}'}{\dd x_{r}} = g_{ir}. \] Hence \begin{align*} (4!)^{3} J^{2} g' &= (4!)^{2} \epsilon_{iklm}\, \epsilon_{rstu}\, g_{ir}\, g_{ks}\, g_{lt}\, g_{mu} \\ &= (4!)^{3} g, \end{align*} which proves the theorem. \PageSep{109} Returning to $E^{\alpha\beta\gamma\delta}$, its tensor-transformation law is \[ E'^{\mu\nu\sigma\tau} = E^{\alpha\beta\gamma\delta}\, \frac{\dd x_{\mu}'}{\dd x_{\alpha}}\, \frac{\dd x_{\nu}'}{\dd x_{\beta}}\, \frac{\dd x_{\sigma}'}{\dd x_{\gamma}}\, \frac{\dd x_{\tau}'}{\dd x_{\delta}}. \] Whence multiplying by~$\epsilon_{\mu\nu\sigma\tau}$ and using~\Eq{(48.1)} \[ E' · \epsilon_{\mu\nu\sigma\tau}\, \epsilon_{\mu\nu\sigma\tau} = E · \epsilon_{\alpha\beta\gamma\delta}\, \epsilon_{\mu\nu\sigma\tau}\, \frac{\dd x_{\mu}'}{\dd x_{\alpha}}\, \frac{\dd x_{\nu}'}{\dd x_{\beta}}\, \frac{\dd x_{\sigma}'}{\dd x_{\gamma}}\, \frac{\dd x_{\tau}'}{\dd x_{\delta}}, \] so that by \Eq{(48.31)} and~\Eq{(48.33)} \[ E' = JE. \Tag{(48.6)} \] Thus $E$~is not an invariant for transformations of coordinates.