We can denote by~$A_{\mu\nu}$ a quantity with $16$~components obtained by giving $\mu$~and $\nu$ the values from~$1$ to~$4$ independently. Similarly $A_{\mu\nu\sigma}$~has $64$~components. \PageSep{52} By a generalisation of the foregoing transformation laws we classify quantities of this kind as follows--- Contravariant tensors \index{Contravariant vectors!tensors}% \begin{align*} A'^{\mu\nu} &= \frac{\dd x_{\mu}'}{\dd x_{\alpha}}\, \frac{\dd x_{\nu}'}{\dd x_{\beta}}\, A^{\alpha\beta}. \Tag{(23.21)} \\ \intertext{\hspace*{\parindent}Covariant tensors} \index{Covariant vector!tensor}% A_{\mu\nu}' &= \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, A_{\alpha\beta}. \Tag{(23.22)} \intertext{\hspace*{\parindent}Mixed tensors} \index{Mixed tensors}% A_{\mu}'^{\nu} &= \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\nu}'}{\dd x_{\beta}}\, A_{\alpha}^{\beta}. \Tag{(23.23)} \end{align*} The above are called tensors of the second rank. We have similar laws for tensors of higher ranks. E.g.\ \[ A_{\mu\nu\sigma}'^{\tau} = \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \frac{\dd x_{\gamma}}{\dd x_{\sigma}'}\, \frac{\dd x_{\tau}'}{\dd x_{\delta}}\, A_{\alpha\beta\gamma}^{\delta}. \Tag{(23.3)} \] It may be worth while to remind the reader that \Eq{(23.3)}~typifies $256$~distinct equations each with a sum of $256$~terms on the right-hand side. It is easily shown that these transformation laws fulfil the condition of self-consistency explained in \SecRef{20}, and it is for this reason that quantities governed by them are selected for special nomenclature. If a tensor vanishes, i.e.\ if all its components vanish, in one system of coordinates, it will continue to vanish when any other system of coordinates is substituted. This is clear from the linearity of the above transformation laws. Evidently the sum of two tensors of the same covariant or contravariant character is a tensor. Hence a law expressed by the vanishing of the sum of a number of tensors, or by the equality of two tensors of the same kind, will be independent of the coordinate-system used. The product of two tensors such as $A_{\mu\nu}$ and $B_{\sigma}^{\tau}$ is a tensor of the character indicated by~$A_{\mu\nu\sigma}^{\tau}$. This is proved by showing that the transformation law of the product is the same as~\Eq{(23.3)}. The general term \emph{tensor} includes vectors (tensors of the first rank) and invariants or scalars\footnote {Scalar is a synonym for invariant. I generally use the latter word as the more self-explanatory.} \index{Scalar}% (tensors of zero rank). A tensor of the second or higher rank need not be expressible as a product of two tensors of lower rank.