We shall define a space-time Vector of the 2nd kind as a system of six-magnitudes f₂₃, f₃₁ ... f₃₄, with the condition that when subjected to a Lorentz transformation, it is changed to a new system f₂₃′ ... f₃₄, ... which satisfies the connection between (23) and (24).
I enunciate in the following manner the general theorem of relativity corresponding to the equations (I)-(iv),—which are the fundamental equations for Äther.
If x, y, z, it (space co-ordinates, and time it) is subjected to a Lorentz transformation, and at the same time (pux, puy, puz, iρ) (convection-current, and charge density ρi) is transformed as a space time vector of the 1st kind, further (mx, my, mz, -iex, -iey, -iez) (magnetic force, and electric induction × (-i) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system.
These facts can be more concisely expressed in these words: the system of equations (I and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where (ρu, iρ) is to be transformed as a space time vector of the 1st kind, (m - ie) is to be treated as a vector of the 2nd kind, or more significantly,—
(ρu, iρ) is a space time vector of the 1st kind, (m - ie)[[17]] is a space-time vector of the 2nd kind.
I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.
(i) m² - e² = f₂₃² + f₃₁² + f₁₂² + f₁₄² + f₂₄² + f₂₄²
me = i(f₂₃f₁₄ + f₃₁f₂₄ + f₁₂f₃₄).
A space-time vector of the second kind (m - ie), where (m and e) are real magnitudes, may be called singular, when the scalar square (m - ie)² = 0, ie m² - e² = 0, and at the same time (m e) = 0, ie the vector m and e are equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every Lorentz-transformation.
If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product [me] coincides with the Z-axis, i.e. mx = 0, ex = 0. Then