I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, _i.e._, instead of operating with (_t_), I shall operate with (_i t_), where _i_ denotes √(-1). If now instead of (_x_, _y_, _z_, _i t_), I use the method of writing with indices, certain essential circumstances will come into evidence; on this will be based a general use of the suffixes (1, 2, 3, 4). The advantage of this method will be, as I expressly emphasize here, that we shall have to handle symbols which have apparently a purely real appearance; we can however at any moment pass to real equations if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities. An individual system of values of (_x_, _y_, _z_, _t_) _i. e._, of (_x₁_ _x₂_ _x₃_ _x₄_) shall be called a space-time point. Further let _u_ denote the velocity vector of matter, ε the dielectric constant, μ the magnetic permeability, σ the conductivity of matter, while ρ denotes the density of electricity in space, and _x_ the vector of “Electric Current” which we shall some across in §7 and §8. PART I § 2. The Limiting Case. The Fundamental Equations for Äther. By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-dynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case ε = 1, μ = 1, σ = 0, they should constitute the laws for ponderable bodies. In this ideal limiting case ε = 1, μ = 1, σ = 0, E will be equal to _e_, and M to _m_. At every space time point (_x_, _y_, _z_, _t_) we shall have the equations[15] (i) Curl _m_ - (δ_e_/δ_t_) = ρu (ii) div _e_ = ρ (iii) Curl _e_ + δ_m_/δ_t_ = 0 (iv) div m = 0 I shall now write (_x₁_ _x₂_ _x₃_ _x₄_) for (_x_, _y_, _z_, _t_) and (ρ₁, ρ₂, ρ₃, ρ₄) for $$ (\rho u_{x}, \rho u_{y}, \rho u_{z}, i\rho) $$ _i.e._ the components of the convection current ρu, and the electric density multiplied by √ -1 Further I shall write _f__{2 3}, _f__{3 1}, _f__{1 2}, _f__{1 4}, _f__{2 4}, _f__{3 4}. for m_{_x_}, m_{_y_}, m_{_z_}, -ie_{_x_}, -ie_{_y_}, -ie_{_z_}. _i.e._, the components of m and (-_i.e._) along the three axes; now if we take any two indices (h. k) out of the series 3, 4), _f__{_k h_} = -_f__{_k h_}, Therefore _f₃₂_ = -_f₂₃_, _f₁₃_ = -_f₃₁_, _f₂₁_ = -_f₁₂_ _f₄₁_ = -_f₁₄_, _f₄₄_ = -_f₂₄_, _f₄₃_ = -_f₃₄_ Then the three equations comprised in (i), and the equation (ii) multiplied by i becomes