$$ (A_{\sigma \tau'} - \frac{\partial x_{\mu}}{\partial x_{\sigma'}}
\frac{\partial x_{\nu}}{\partial x_{\tau'}}) B^{\sigma \tau'} = 0 $$
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.