Consider a tensor of the fourth rank $E^{\alpha\beta\gamma\delta}$ which is antisymmetrical for all pairs of suffixes. Any component with two suffixes alike must be zero, since by the rule of antisymmetry $E^{\alpha\beta11} = -E^{\alpha\beta11}$. In the surviving components, $\alpha$, $\beta$, $\gamma$, $\delta$, being all different, must stand for the numbers $1$, $2$, $3$, $4$ in arbitrary order. We can pass from any of these components to~$E^{1234}$ by a series of interchanges of the suffixes in pairs, and each interchange merely reverses the sign. Writing $E$ for~$E^{1234}$, all the $256$~components have one or other of the values \[ +E,\quad 0,\quad -E. \] We shall write \[ E^{\alpha\beta\gamma\delta} = E · \epsilon_{\alpha\beta\gamma\delta}, \Tag{(48.1)} \] where \begin{align*} \epsilon_{\alpha\beta\gamma\delta} &= \Neg0, \text{ when the suffixes are not all different,} \\ &= +1, \parbox[t]{0.8\textwidth}{ when they can be brought to the order $1$, $2$, $3$, $4$ by an even number of interchanges,} \\ &= -1, \text{ when an odd number of interchanges is needed.} \end{align*} It will appear later that $E$~is not an invariant; consequently $\epsilon_{\alpha\beta\gamma\delta}$ is not a tensor. The coefficient $\epsilon_{\alpha\beta\gamma\delta}$ is particularly useful for dealing with determinants. If $|k_{\mu\nu}|$ denotes the determinant formed with the elements~$k_{\mu\nu}$ (which need \index{Determinants, manipulation of}% not form a tensor), we have \[ 4! \times |k_{\mu\nu}| = \epsilon_{\alpha\beta\gamma\delta} \epsilon_{\epsilon\zeta\eta\theta}\, k_{\alpha\epsilon} k_{\beta\zeta} k_{\gamma\eta} k_{\delta\theta}, \Tag{(48.2)} \] because the terms of the determinant are obtained by selecting four elements, one from each row ($\alpha$, $\beta$, $\gamma$, $\delta$, all different) and also from each column ($\epsilon$, $\zeta$, $\eta$, $\theta$, all different) and affixing the $+$~or $-$ sign to the product according as the order of the columns is brought into the order of the rows by an even or odd number of interchanges. The factor~$4!$ appears because every possible permutation of the same four elements is included separately in the summation on the right. It is possible by corresponding formulae to define and manipulate determinants in three dimensions (with $64$~elements arranged in a cube) or in four dimensions. Note that \[ \epsilon_{\alpha\beta\gamma\delta} \epsilon_{\epsilon\zeta\eta\theta} = 4!. \Tag{(48.31)} \] %\PageSep{108}