The determinants with which we are most concerned are the fundamental determinant~$g$ and the Jacobian of a transformation \index{Jacobian}% \[ J = \frac{\dd(x_{1}', x_{2}', x_{3}', x_{4}')}{\dd(x_{1}, x_{2}, x_{3}, x_{4})}. \] By~\Eq{(48.2)} \begin{align*} 4!\, g &= \epsilon_{\alpha\beta\gamma\delta}\, \epsilon_{\epsilon\zeta\eta\theta}\, g_{\alpha\epsilon}\, g_{\beta\zeta}\, g_{\gamma\eta}\, g_{\delta\theta}, \Tag{(48.32)} \\ 4!\, J &= \epsilon_{\alpha\beta\gamma\delta}\, \epsilon_{\epsilon\zeta\eta\theta}\, \frac{\dd x_{\epsilon}'}{\dd x_{\alpha}}\, \frac{\dd x_{\zeta}'}{\dd x_{\beta}}\, \frac{\dd x_{\eta}'}{\dd x_{\gamma}}\, \frac{\dd x_{\theta}'}{\dd x_{\delta}}. \Tag{(48.33)} \end{align*} To illustrate the manipulations we shall prove that\footnote {A shorter proof is given at the end of this section.} \[ g = J^{2} g'. \] By \Eq{(48.32)} and~\Eq{(48.33)} \begin{multline*} (4!)^{3} J^{2} g' = \epsilon_{\alpha\beta\gamma\delta}\, \epsilon_{\epsilon\zeta\eta\theta}\, g_{\alpha\epsilon}'\, g_{\beta\zeta}'\, g_{\gamma\eta}'\, g_{\delta\theta}' · \epsilon_{\iota\kappa\lambda\mu}\, \epsilon_{\nu\xi\omicron\varpi}\, \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_{\rho\sigma\tau\upsilon}\, \epsilon_{\phi\chi\psi\omega}\, \frac{\dd x_{\phi}'}{\dd x_{\rho}}\, \frac{\dd x_{\chi}'}{\dd x_{\sigma}}\, \frac{\dd x_{\psi}'}{\dd x_{\tau}}\, \frac{\dd x_{\omega}'}{\dd x_{\upsilon}}. \Tag{(48.41)} \end{multline*} There are about $280$~billion terms on the right, and we proceed to rearrange those which do not vanish. For non-vanishing terms the letters $\nu$, $\xi$, $\omicron$, $\varpi$ denote the same suffixes as $\alpha$, $\beta$, $\gamma$, $\delta$, but (usually) in a different order. Permute the four factors in which they occur so that they come into the same order; the suffixes of the denominators will then come into a new order, say, $i$, $k$, $l$,~$m$. Thus \[ \frac{\dd x_{\nu}'}{\dd x_{\iota}}\, \frac{\dd x_{\xi}'}{\dd x_{\kappa}}\, \frac{\dd x_{\omicron}'}{\dd x_{\lambda}}\, \frac{\dd x_{\varpi}'}{\dd x_{\mu}} = \frac{\dd x_{\alpha}'}{\dd x_{i}}\, \frac{\dd x_{\beta}'}{\dd x_{k}}\, \frac{\dd x_{\gamma}'}{\dd x_{l}}\, \frac{\dd x_{\delta}'}{\dd x_{m}}. \Tag{(48.42)} \]