ω₁ = u_{_x_}/√(1 - _u²_), ω₂ = u_{_y_}/√(1 - u²), ω₃ = u_{_z_}/√(1 - u²), ω₄ = _i_/√(1 - u²) with the relation (27) ω₁² + ω₂² + ω₃² + ω₄² = - | From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space-time vector of the 1st kind. Let us now fix our attention on a certain point (_x_, _y_, _z_) of matter at a certain time (_t_). If at this space-time point _u_ = 0, then we have at once for this point the equations (_A_), (_B_) (_V_) of § 7. If _u_ ≠ 0, then there exists according to 16), in case | _u_ | < 1, a special Lorentz-transformation, whose vector _v_ is equal to this vector _u_ (_x_, _y_, _z_, _t_), and we pass on to a new system of reference (_x′_ _y′_ _z′_ _t′_) in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = _i_, therefore the new velocity vector ω′ = 0, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point (_x′_ _y′_ _z′_ _t_) involves the newly introduced magnitude (_u′_ ρ′, C′, _e′_, _m′_, _E′_, _M′_) and their differential quotients with respect to (_x′_, _y′_, _z′_, _t′_) in the same manner as the original equations for the point (_x_, _y_, _z_, _t_). But according to the first axiom, when _u′_ = 0, these equations must be exactly equivalent to (1) the differential equations (_A′_), (_B′_), which are obtained from the equations (_A_), (_B_) by simply dashing the symbols in (_A_) and (_B_). (2) and the equations (V′) _e′_ = ε_E′_, _M’_ = μ_m′_, _C′_ = σ_E′_ where ε, μ, σ are the dielectric constant, magnetic permeability, and conductivity for the system (_x′_ _y′_ _z′_ _t′_) _i.e._ in the space-time point (_x_ _y_, _z_ _t_) of matter. Now let us return, by means of the reciprocal Lorentz-transformation to the original variables (_x_, _y_, _z_, _t_), and the magnitudes (_u_, ρ, C, _e_, _m_, _E_, _M_) and the equations, which we then obtain from the last mentioned, will be the fundamental equations sought by us for the moving bodies. Now from § 4, and § 6, it is to be seen that the equations _A_), as well as the equations _B_) are covariant for a Lorentz-transformation, _i.e._ the equations, which we obtain backwards from _A′_) _B′_), must be exactly of the same form as the equations _A_) and _B_), as we take them for bodies at rest. We have therefore as the first result:— The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in ρ and the vectors C, _e_, _m_, E, M, are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have