Should √(-_g_) vanish at any point of the four-dimensional continuum it would signify that to a finite co-ordinate volume at the place corresponds an infinitely small “natural volume.” This can never be the case; so that _g_ can never change its sign; we would, according to the special relativity theory assume that _g_ has a finite negative value. It is a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice of co-ordinates. If however (-_g_) remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one. We would afterwards see that such a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature. In place of (18) it follows then simply that _d_τ′ = _d_ from this it follows, remembering the law of Jacobi, (19) $$ | \frac{\partial x'_{\sigma}}{dx_{\mu}} | = 1 $$ With this choice of co-ordinates, only substitutions with determinant 1 are allowable. It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate. We do not seek those laws of nature which are co-variants with regard to the transformations having the determinant 1, but we ask: what are the general co-variant laws of nature? First we get the law, and then we simplify its expression by a special choice of the system of reference. _Building up of new tensors with the help of the fundamental tensor._ Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed. Example:— A^{μ} = _g_^{μσ} A_{σ} A = _g__{μν} A^{μν} We would point out specially the following combinations: A^{μν} = _g_^{μα} _g_^{νβ} A_{αβ} A_{μν} = _g__{μα} _g__{νβ} A^{αβ} (complement to the co-variant or contravariant tensors) and B_{μν} = _g__{μν} _g_^{αβ} A_{αβ} We can call B_{μν} the reduced tensor related to A_{μν}. Similarly B^{μν} = _g_^{μν}_g__{αβ}A^{αβ}. It is to be remarked that _g_^{μν} is no other than the “complement” of _g__{μν} for we have,— _g_^{μα}_g_^{νβ}_g__{αβ} = _g__{μα}δ^{ν}_{α} = _g_^{μν}. § 9. Equation of the geodetic line (or of point-motion). As the “line element” _ds_ is a definite magnitude independent of the co-ordinate system, we have also between two points P₁ and P₂ of a four dimensional continuum a line for which ∫_ds_ is an extremum (geodetic line), _i.e._, one which has got a significance independent of the choice of co-ordinates. Its equation is (20) δ{ ∫^{P₂}_{P₁} _ds_ } = 0