_g__{μσ} _g__{ντ} _g_^{στ} _dx__{μ} _dx__{ν} Now according to the rules of multiplication, of the fore-going paragraph, the magnitudes _d_ξ_{σ} = _g__{μσ} _dx__{μ} forms a co-variant four-vector, and in fact (on account of the arbitrary choice of _dx__{μ}) any arbitrary four-vector. If we introduce it in our expression, we get _ds²_ = _g_^{στ} _d_ξ_{σ} _d_ξ_{τ}. For any choice of the vectors _d_ξ_{σ} _d_ξ_{τ} this is scalar, and _g_^{στ}, according to its definition is a symmetrical thing in σ and τ, so it follows from the above results, that _g_^{στ} is a contravariant tensor. Out of (16) it also follows that δ^{ν}_{μ} is a tensor which we may call the mixed fundamental tensor. _Determinant of the fundamental tensor._ According to the law of multiplication of determinants, we have | _g__{μα} _g_^{αν} | = | _g__{μα} | | _g_^{αν} | On the other hand we have | _g__{μα} _g_^{αν} | = | δ^{ν}_{μ} | = 1 So that it follows (17) that | _g__{μν} | | _g_^{μν} | = 1. _Invariant of volume._ We see first the transformation law for the determinant _g_ = | _g__{μν} |. According to (11) $$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial x_{\nu}}{\partial x_{\tau'}} g_{\mu u} | $$ From this by applying the law of multiplication twice, we obtain $$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} | | \frac{\partial x_{\nu}}{\partial x_{\tau'}} | | g_{\mu \nu} | = | \frac{\partial x_{\mu}}{\alpha_{\sigma'}} | g $$ or (A) $$ \sqrt{g'} = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} | \sqrt{g} $$ On the other hand the law of transformation of the volume element _d_τ′ = ∫ _dx₁_ _dx₂_ _dx₃_ _dx₄_ is according to the wellknown law of Jacobi. (B) $$ d\tau' = | \frac{dx'_{\sigma}}{dx_{\mu}} | d\tau $$ by multiplication of the two last equations (A) and (B) we get (18) = √_g_ _d_τ′ = √_g_ _d_τ. Instead of √_g_, we shall afterwards introduce √(-_g_) which has a real value on account of the hyperbolic character of the time-space continuum. The invariant √(-_g_)_d_τ, is equal in magnitude to the four-dimensional volume-element measured with solid rods and clocks, in accordance with the special relativity theory. _Remarks on the character of the space-time continuum_—Our assumption that in an infinitely small region the special relativity theory holds, leads us to conclude that _ds²_ can always, according to (1) be expressed in real magnitudes _d_X₁ ... _d_X_{_h_}. If we call _d_τ₀ the “_natural_” volume element _d_X₁ _d_X₂ _d_X₃ _d_X₄ we have thus (18a) _d_τ₀ = √(_g_)_i_τ.