This can be true, for any choice of B^{στ′} only when the term within the bracket vanishes. From which by referring to (11), the theorem at once follows. This law correspondingly holds for tensors of any rank and character. The proof is quite similar. The law can also be put in the following form. If B^{μ} and C^{ν} are any two vectors, and if for every choice of them the inner product A_{μν} B^{μ} C^{ν} is a scalar, then A_{μν} is a co-variant tensor. The last law holds even when there is the more special formulation, that with any arbitrary choice of the four-vector B^{μ} alone the scalar product A_{μν} B^{μ} B^{ν} is a scalar, in which case we have the additional condition that A_{μν} satisfies the symmetry condition. According to the method given above, we prove the tensor character of (A_{μν} + A_{νμ}), from which on account of symmetry follows the tensor-character of A_{μν}. This law can easily be generalized in the case of co-variant and contravariant tensors of any rank. Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind of tensor: If the quantities A_{μν} B^{ν} form a tensor of the first rank, when B^{ν} is any arbitrarily chosen four-vector, then A_{μν} is a tensor of the second rank. If for example, C^{μ} is any four-vector, then owing to the tensor character of A_{μν} B^{ν}, the inner product A_{μν} C^{μ} B^{ν} is a scalar, both the four-vectors C^{μ} and B^{ν} being arbitrarily chosen. Hence the proposition follows at once. A few words about the Fundamental Tensor _g__{μν}. The co-variant fundamental tensor—In the invariant expression of the square of the linear element _ds²_ = _g__{μν} _dx__{μ} _dx__{ν} _dx__{μ} plays the rôle of any arbitrarily chosen contravariant vector, since further _g__{μν} = _g__{νμ}, it follows from the considerations of the last paragraph that _g__{μν} is a symmetrical co-variant tensor of the second rank. We call it the “fundamental tensor.” Afterwards we shall deduce some properties of this tensor, which will also be true for any tensor of the second rank. But the special rôle of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravitation makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor. _The co-variant fundamental tensor._ If we form from the determinant scheme | _g__{μν} | the minors of _g__{μν} and divide them by the determinant _g_ = | _g__{μν} | we get certain quantities _g_^{μν} = _g_^{νμ}, which as we shall prove generates a contravariant tensor. According to the well-known law of Determinants (16) _g__{μσ} _g_^{νσ} = δ_{μ}^{ν} where δ_{μ}^{ν} is 1, or 0, according as μ = ν or not. Instead of the above expression for _ds²_, we can also write _g__{μσ} δ_{ν}^{σ} _dx__{μ} _dx__{ν} or according to (16) also in the form