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.