The first Maxwellian system is obtained by a generalisation of the form given by Minkowski.
We introduce the contra-variant six-vector F_{αβ} by the equation
(62) F^{μν} = g^{μα} g^{νβ} F_{αβ},
and also a contra-variant four-vector J^μ, which is the electrical current-density in vacuum. Then remembering (40) we can establish the system of equations, which remains invariant for any substitution with determinant 1 (according to our choice of co-ordinates).
(63) [part]F^{μν}/[part]x_{ν} = J^μ
If we put
{ F^{2 3} = H´_{x} F^{1 4} = -E´_{x}
{
(64) { F^{3 1} = H´_{y} F^{2 4} = -E´_{y}
{
{ F^{1 2} = H´_{z} F^{3 4} = -E´_{z}
which quantities become equal to H_{x} . . . E_{x} in the case of the special relativity theory, and besides
J^1 = i_{x} . . . J^4 = ρ
we get instead of (63)
{ rot H´ - [part]E´/[part]t = i
(63a) {
{ div E´ = ρ
