_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
$$ \begin{vmatrix} & \frac{\delta f_{1 2}}{\delta x_{2}} & +
\frac{\delta f_{1 3}}{\delta x_{3}} & + \frac{\delta f_{1 4}}{\delta
x_{4}} & = \rho_{1} \frac{\delta f_{2 1}}{\delta x_{1}} & & +
\frac{\delta f_{2 3}}{\delta x_{3}} & \times \frac{\delta f_{2
4}}{\delta x_{4}} & = \rho_{2} \frac{\delta f_{3 1}}{\delta x_{1}} &
\times \frac{\delta f_{3 2}}{\delta x_{2}} & & + \frac{\delta f_{3
4}}{\delta x_{4}} & = \rho_{3} \frac{\delta f_{4 1}}{\delta x_{1}} & +
\frac{\delta f_{4 2}}{\delta x_{2}} & + \frac{\delta f_{4 3}}{\delta
x_{3}} & & = \rho_{4} \end{vmatrix} × $$
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
_x₁′_ = _x₁_,
_x₂′_ = _x₂_,
_x₃′_ = _x₃_ cos _i_ψ + _x₄_ sin _i_ψ, (1)
_x₄′_´ = - _x₃_ sin _i_ψ + _x₄_ cos _i_ψ,