The three last equations which have been just cited here coincide with eqn (II), (III), (IV), the first equation would be, if J is identified with C, = uρ (the current being zero for σ = 0,
(29) Curl [H - (u, E)] = C + dD/dt - curl [uD],
and thus comes out to be in a different form than (1) here. Therefore for magnetised bodies, Lorentz’s equations do not correspond to the Relativity Principle.
On the other hand, the form corresponding to the relativity principle, for the condition of non-magnetisation is to be taken out of (D) in [§8], with μ = 1, not as B = H, as Lorentz takes, but as (30) B - [uD] = H - [uD] (M - [uE] = m - [ue]. Now by putting H = B, the differential equation (29) is transformed into the same form as eqn (1) here when m - [ue] = M - [uE]. Therefore it so happens that by a compensation of two contradictions to the relativity principle, the differential equations of Lorentz for moving non-magnetised bodies at last agree with the relativity postulate.
If we make use of (30) for non-magnetic bodies, and put accordingly H = B + [u, (D - E)], then in consequence of (C) in [§8],
(ε - 1) (E + [u, B]) = D - E + [u. [u, D - E],
i.e. for the direction of u,
(ε - 1) (E + [uB])u = (D - E)u
and for a perpendicular direction ū,
(ε - 1) [E + (uB)]u = (1 - u²) (D - E)u