(16) _g__{μσ} _g_^{νσ} = δ_{μ}^{ν}
where δ_{μ}^{ν} is 1, or 0, according as μ = ν or not. Instead of the
above expression for _ds²_, we can also write
_g__{μσ} δ_{ν}^{σ} _dx__{μ} _dx__{ν}
or according to (16) also in the form
_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.