_Mixed tensor._ We can also define a tensor of the second rank of the type (12) A_{μ}^{ν} = A_{μ}B^{ν} which is co-variant with reference to μ and contravariant with reference to ν. Its transformation law is (13) $$ A^{\tau'}_{\sigma} = \frac{\partial x_{\tau'}}{\partial x_{\beta}} \frac{\partial \alpha}{\partial x_{\sigma'}} A^{\beta}_{\alpha} $$ Naturally there are mixed tensors with any number of co-variant indices, and with any number of contra-variant indices. The co-variant and contra-variant tensors can be looked upon as special cases of mixed tensors. _Symmetrical tensors_:— A contravariant or a co-variant tensor of the second or higher rank is called symmetrical when any two components obtained by the mutual interchange of two indices are equal. The tensor A^{μν} or A_{μν} is symmetrical, when we have for any combination of indices (14) A^{μν} = A^{νμ} or (14a) A_{μν} = A_{νμ}. It must be proved that a symmetry so defined is a property independent of the system of reference. It follows in fact from (9) remembering (14) $$ A^{\sigma \tau'} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x_{\tau'}}{\partial x_{\nu}} A^{\nu \mu} = A^{\tau \sigma'} $$ _Antisymmetrical tensor._ A contravariant or co-variant tensor of the 2nd, 3rd or 4th rank is called _antisymmetrical_ when the two components got by mutually interchanging any two indices are equal and opposite. The tensor or A^{μν} or A_{μν} is thus antisymmetrical when we have (15) A^{μν} = -A^{νμ} or (15a) A_{μν} = -A_{νμ}. Of the 16 components A^{μν}, the four components A^{μμ} vanish, the rest are equal and opposite in pairs; so that there are only 6 numerically different components present (Six-vector). Thus we also see that the antisymmetrical tensor A^{μνσ} (3rd rank) has only 4 components numerically different, and the antisymmetrical tensor A^{μνστ} only one. Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions. § 7. Multiplication of Tensors. _Outer multiplication of Tensors_:—We get from the components of a tensor of rank _z_, and another of a rank _z′_, the components of a tensor of rank (_z_ + _z′_) for which we multiply all the components of the first with all the components of the second in pairs. For example, we obtain the tensor Τ from the tensors A and B of different kinds:— Τ_{μνσ} = A_{μν}B_{σ}, Τ^{αβγδ} = A^{αβ}B^{γδ}, Τ_{αβ}^{γδ} = A_{αβ}B^{γδ}.