_Remarks on the simplification of the mode of writing the expressions._ A glance at the equations of this paragraph will show that the indices which appear twice within the sign of summation [for example ν in (5)] are those over which the summation is to be made and that only over the indices which appear twice. It is therefore possible, without loss of clearness, to leave off the summation sign; so that we introduce the rule: wherever the index in any term of an expression appears twice, it is to be summed over all of them except when it is not expressedly said to the contrary. The difference between the co-variant and the contra-variant four-vector lies in the transformation laws [(7) and (5)]. Both the quantities are tensors according to the above general remarks; in it lies its significance. In accordance with Ricci and Levi-civita, the contravariants and co-variants are designated by the over and under indices. § 6. Tensors of the second and higher ranks. Contravariant tensor:—If we now calculate all the 16 products A^{μν} of the components A^{μ} B^{ν}, of two contravariant four-vectors (8) A^{μν} = A^{μ}B^{ν} A^{μν}, will according to (8) and (5 a) satisfy the following transformation law. (9) $$ A^{\sigma \tau'} = \frac{\partial x'_{\sigma}}{\partial x_{\mu}} \frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} $$ We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transformation relation (9), a contravariant tensor of the second rank. Not every such tensor can be built from two four-vectors, (according to 8). But it is easy to show that any 16 quantities A^{μν}, can be represented as the sum of A^{μ}B^{ν} of properly chosen four pairs of four-vectors. From it, we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through (9), by proving it only for the special tensor of the type (8). _Contravariant Tensor of any rank_:—It is clear that corresponding to (8) and (9), we can define contravariant tensors of the 3rd and higher ranks, with 4³, etc. components. Thus it is clear from (8) and (9) that in this sense, we can look upon contravariant four-vectors, as contravariant tensors of the first rank. _Co-variant tensor._ If on the other hand, we take the 16 products A_{μν} of the components of two co-variant four-vectors A_{μ} and B_{ν}, (10) A_{μν} = A_{μ} B_{ν}. for them holds the transformation law (11) $$ A^{\sigma \tau'} = \frac{\partial x'_{\mu}}{\partial x_{\sigma'}} \frac{\partial x'_{\nu}}{\partial x_{\tau'}} A^{\mu \nu} $$ By means of these transformation laws, the co-variant tensor of the second rank is defined. All re-marks which we have already made concerning the contravariant tensors, hold also for co-variant tensors. _Remark_:— It is convenient to treat the scalar Invariant either as a contravariant or a co-variant tensor of zero rank.