Page:The principle of relativity (1920).djvu/102

 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] in book, so not µ on top of D] (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 - u^2) (D - E)_{u}]

i.e. it coincides with Lorentz's assumption, if we neglect u2 in comparison to 1.

Also to the same order of approximation, Lorentz's form for J corresponds to the conditions imposed by the relativity principle [comp. (E) § 8]—that the components of J_{u}, J_{[=u]} are equal to the components of σ (E + [u B]) multiplied by [sqrt](1 - u^2) or 1 / [sqrt](1 - u^2) respectively.