and write _s₁_, _s₂_, _s₃_, _s₄_ for C_{_x_}, C_{_y_}, C_{_z_} (√-1)ρ. Further _f₂₃_, _f₃₁_, _f₁₂_, _f₁₄_, _f₂₄_, _f₃₄_ for _m__{_x_}, _m__{_y_}, _m__{_z_}, -_i_(_e__{_x_}, _e__{_y_}, _e__{_z_}), and F₂₃, F₃₁, F₁₂, F₁₄, F₂₄, F₃₄ for M_{_x_}, M_{_y_}, M_{_z_}, -_i_(E_{_x_}, E_{_y_}, E_{_z_}) lastly we shall have the relation _f__{k h} = - _f__{_h k_}, _F__{_k h_} = -_F__{_h k_}, (the letter _f_, F shall denote the field, _s_ the (_i.e._ current). Then the fundamental Equations can be written as (A) ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = s₁ ∂_f₂₁_/∂_x₁_ + + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = s₂ ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + + ∂_f₃₄_/∂_x₄_ = s₃ ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = s₄ and the equations (3) and (4), are ∂F₃₄/∂_x₂_ + ∂F₄₂/∂_x₃_ + ∂F₂₃/∂_x₄_ = 0 ∂F₄₃/∂_x₁_ + + ∂F₁₄/∂_x₃_ + ∂F₃₁∂_x₄_ = 0 ∂F₂₄/∂_x₁_ + ∂F₄₁/∂_x₂_ + + ∂F₁₂/∂_x₄_ = 0 ∂F₃₂/∂_x₁_ + ∂F₁₃/∂_x₂_ + ∂F₂₁/∂_x₃_ = 0 § 8. The Fundamental Equations. We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively. The first Axion shall be,— When a detached region[19] of matter is at rest at any moment, therefore the vector _u_ is zero, for a system (_x_, _y_, _z_, _t_)—the neighbourhood may be supposed to be in motion in any possible manner, then for the space-time point _x_, _y_, _z_, _t_, the same relations (A) (B) (V) which hold in the case when all matter is at rest, shall also hold between ρ, the vectors C, _e_, _m_, _M_, _E_ and their differentials with respect to _x_, _y_, _z_, _t_. The second axiom shall be:— Every velocity of matter is < 1, smaller than the velocity of propagation of light.[20] The fundamental equations are of such a kind that when (_x_, _y_, _z_, _it_) are subjected to a Lorentz transformation and thereby (_m_ - _ie_) and (_M_ - _iE_) are transformed into space-time vectors of the second kind, (C, _i_ρ) as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes. Shortly I can signify the third axiom as:— (_m_, -_ie_), and (_M_, -_iE_) are space-time vectors of the second kind, (C, _i_p) is a space-time vector of the first kind. This axiom I call the Principle of Relativity. In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way. According to the second axiom, the magnitude of the velocity vector | _u_ | is < 1 at any space-time point. In consequence, we can always write, instead of the vector _u_, the following set of four allied quantities