$$ B^{\rho}_{\mu\sigma\tau} = - \frac{\partial}{\partial x_{\tau}} \begin{Bmatrix}\mu & & \sigma\ \rho&\end{Bmatrix} + \frac{\partial}{\partial x_{\sigma}} \begin{Bmatrix}\mu & & \tau\ \rho&\end{Bmatrix} - \begin{Bmatrix}\mu & & \sigma\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & & \tau\ \rho&\end{Bmatrix} + \begin{Bmatrix}\mu & & \tau\ \alpha&\end{Bmatrix} \begin{Bmatrix}\alpha & & \sigma\ \rho&\end{Bmatrix} $$ The essential thing in this result is that on the right hand side of (42) we have only A_{ρ}, but not its differential co-efficients. From the tensor-character of A_{μστ} - A_{μτσ}, and from the fact that A_{ρ} is an arbitrary four vector, it follows, on account of the result of §7, that B^{ρ}_{μστ} is a tensor (Riemann-Christoffel Tensor). The mathematical significance of this tensor is as follows; when the continuum is so shaped, that there is a co-ordinate system for which _g__{μν}_’s_ are constants, B^{ρ}_{μστ} all vanish. If we choose instead of the original co-ordinate system any new one, so would the _g__{μν}’s referred to this last system be no longer constants. The tensor character of B^{ρ}_{μστ} shows us, however, that these components vanish collectively also in any other chosen system of reference. The vanishing of the Riemann Tensor is thus a necessary condition that for some choice of the axis-system _g__{μν}’s can be taken as constants. In our problem it corresponds to the case when by a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region. By the reduction of (43) with reference to indices to τ and ρ, we get the covariant tensor of the second rank (44) $$ B_{\mu\nu} = R_{\mu\nu} + S_{\mu\nu} $$ $$ R_{\mu\nu} = - \frac{\partial}{\partial x_{\alpha}} \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} + \begin{Bmatrix}\mu & & \alpha\ \beta&\end{Bmatrix} \begin{Bmatrix}\nu & & \beta\ \alpha&\end{Bmatrix} $$ $$ S_{\mu\nu} = \frac{\partial \log \sqrt{-g}}{\partial x_{\mu} \partial x_{\nu}} - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} \frac{\partial \log \sqrt{-g}}{\partial x_{\alpha}} $$ _Remarks upon the choice of co-ordinates._—It has already been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so chosen that √(-_g_) = 1. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplification. It is specially true for the tensor B_{μν}, which plays a fundamental rôle in the theory. By this simplification, S_{μν} vanishes of itself so that tensor B_{μν} reduces to R_{μν}. I shall give in the following pages all relations in the simplified form, with the above-named specialisation of the co-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any special case. C. THE THEORY OF THE GRAVITATION-FIELD