$$ \begin{vmatrix} & \frac{\delta f_{1 2}}{\delta x_{2}} & + \frac{\delta f_{1 3}}{\delta x_{3}} & + \frac{\delta f_{1 4}}{\delta x_{4}} & = \rho_{1} \frac{\delta f_{2 1}}{\delta x_{1}} & & + \frac{\delta f_{2 3}}{\delta x_{3}} & \times \frac{\delta f_{2 4}}{\delta x_{4}} & = \rho_{2} \frac{\delta f_{3 1}}{\delta x_{1}} & \times \frac{\delta f_{3 2}}{\delta x_{2}} & & + \frac{\delta f_{3 4}}{\delta x_{4}} & = \rho_{3} \frac{\delta f_{4 1}}{\delta x_{1}} & + \frac{\delta f_{4 2}}{\delta x_{2}} & + \frac{\delta f_{4 3}}{\delta x_{3}} & & = \rho_{4} \end{vmatrix} × $$ On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by (_i_) becomes $$ \begin{vmatrix} & \frac{\delta f_{3 4}}{\delta x_{2}} & + \frac{\delta f_{4 2}}{\delta x_{3}} & + \frac{\delta f_{2 3}}{\delta x_{4}} & = = \frac{\delta f_{4 3}}{\delta x_{1}} & & + \frac{\delta f_{1 4}}{\delta x_{3}} & + \frac{\delta f_{3 1}}{\delta x_{4}} & = 0 \frac{\delta f_{2 4}}{\delta x_{1}} & + \frac{\delta f_{4 1}}{\delta x_{2}} & & + \frac{\delta f_{1 2}}{\delta x_{4}} & = 0 \frac{\delta f_{3 2}}{\delta x_{1}} & + \frac{\delta f_{1 3}}{\delta x_{2}} & + \frac{\delta f_{2 1}}{\delta x_{3}} & & = - \end{vmatrix} × $$ By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices, (1, 2, 3, 4). § 3. It is well-known that by writing the equations i) to iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equations A) as well as of B), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis, through an amount φ, keeping e, m fixed in space, and introduce new variables _x₁′_ _x₂′_ _x₃′_ _x₄′_ instead of _x₁_ _x₂_ _x₃_ _x₄_ where _x′₁_ = _x₁_ cos φ + _x₂_ sin φ, _x′₂_ = -_x₁_ sin φ + _x₂_ cos φ, _x′₃_ = _x₃_, _x′₄_ = _x₄_, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ and _f′__{1 2}, ... ... _f′__{3 4}, where _f′₂₃_ = _f₂₃_ cos φ + _f₃₁_ sin φ, _f′₃₁_ = - _f₂₃_ sin φ + _f₃₁_ cos φ, _f′₁₂_ = _f₁₂_, _f′₁₄_ = _f₁₄_ cos φ + _f₂₄_ sin φ, _f′₂₄_ = - _f₁₄_ sin φ + _f₂₄_ cos φ, _f′₃₄_ = _f₃₄__{3 4}, _f′__{_k h_} = - _f__{_k h_} (h l k = 1, 2, 3, 4). then out of the equations (A) would follow a corresponding system of dashed equations (A´) composed of the newly introduced dashed magnitudes. So upon the ground of symmetry alone of the equations (A) and (B) concerning the _suffixes_ (1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all. I will denote by _i_ψ a purely imaginary magnitude, and consider the substitution