are tensors. Through outer multiplication of the first with B_{ν} and the 2nd with A_{μ} we get tensors of the third rank. Their addition gives the tensor of the third rank (27) $$ A_{\mu\nu\sigma} = \frac{\partial A_{\mu\nu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau\nu} - \begin{Bmatrix}\sigma\nu\\\tau\end{Bmatrix} A_{\mu\tau} $$ where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear and homogeneous with reference to A_{μν}, and its first differential co-efficient, so that this law of formation leads to a tensor not only in the case of a tensor of the type A_{μ} B_{ν} but also in the case of a summation for all such tensors, _i.e._, in the case of any co-variant tensor of the second rank. We call A_{μνσ} the extension of the tensor A_{μν}. It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the first and zero rank). In general we can get all special laws of formation of tensors from (27) combined with tensor multiplication. Some special cases of Particular Importance. _A few auxiliary lemmas concerning the fundamental tensor._ We shall first deduce some of the lemmas much used afterwards. According to the law of differentiation of determinants, we have (28) _dg_ = _g_^{μν} _g dg__{μν} = -_g__{μν} _gdg_^{μν}. The last form follows from the first when we remember that _g__{μν} _g_^{μ′ν} = δ^{μ′}_{μ} , and therefore _g__{μν}_g_^{μν} = 4, consequently _g__{μν}_dg_^{μν} + _g_^{μν} _dg__{μν} = 0. From (28), it follows that (29) $$ \frac{1}{\sqrt{-g}} \frac{\partial \sqrt{-g}}{\partial x_{\sigma}} = \frac{1}{2} \frac{\log (-g)}{\partial x_{\sigma}} = \frac{1}{2} g^{\mu\nu} \frac{\partial g_{\mu\nu}}{\partial x_{\sigma}} = - \frac{1}{2} g_{\mu\nu} \frac{\partial g^{\mu\nu}}{\partial x_{\sigma}} $$ Again, since _g__{μν} _g_^{νσ} = δ^{ν}_{μ} , we have, by differentiation, $$ g_{\mu\sigma} dg^{\nu\sigma} = -g^{\nu\sigma} dg_{\mu\sigma} $$ or $$ g_{\mu\sigma} \frac{\partial g^{\nu\sigma}}{\partial x_{\lambda}} = - g^{\nu\sigma} \frac{\partial g_{\mu\sigma}}{\partial x_{\lambda}} $$ By mixed multiplication with _g_^{στ} and _g__{νλ} respectively we obtain (changing the mode of writing the indices). (31) _dg_^{μν} = -_g_^{μα} _g_^{νβ} _dg__{αβ} ∂_g_^{μν}/∂_x__{σ} = -_g_^{μα} _g_^{νβ} _dg__{αβ} and (32) _dg__{μν} = -_g__{μα} _g__{νβ} _dg_^{αβ} ∂_g__{μν}/∂_x__{σ} = -_g__{μα} _g__{νβ} ∂_g_^{αβ}/∂_x__{σ}. The expression (31) allows a transformation which we shall often use; according to (21) (33) $$ \frac{\partial g_{\alpha\beta}}{\partial x_{\sigma}} = \begin{bmatrix}\alpha & & \sigma\ & \beta &\end{bmatrix} + \begin{bmatrix}\beta & & \sigma\ \alpha&\end{bmatrix} $$ If we substitute this in the second of the formula (31), we get, remembering (23), (34)