The velocity of matter occurs only in the auxiliary equations which
characterise the influence of matter on the basis of their
characteristic constants ε, μ, σ. Let us now transform these auxiliary
equations V′) into the original co-ordinates (_x_, _y_, _z_, and _t_.)
According to formula 15) in § 4, the component of _e′_ in the direction
of the vector _u_ is the same as that of (_e_ + [_u_ _m_]), the
component of _m′_ is the same as that of _m_ - [_u_ _e_], but for the
perpendicular direction _ū_, the components of _e′_, _m′_ are the same
as those of (_e_ + [_u_ _m_]) and (_m_ - [_u_ _e_], multiplied by 1/√(1
- _u²_). On the other hand E′ and M′ shall stand to E + [_u_M], and M -
[_u_E] in the same relation as _e′_ and _m′_ to _e_ + [_um_], and _m_ -
(_ue_). From the relation _e′_ = εE′, the following equations follow
(C) _e_ + [_um_] = ε(E + [_u_M]),
and from the relation M′ = μ_m′_, we have
(D) M - [_u_ E] = μ(_m_ - [_ue_]),
For the components in the directions perpendicular to _u_, and to each
other, the equations are to be multiplied by √(1 - _u²_).
Then the following equations follow from the transformation? equations
(12), (10), (11) in § 4, when we replace q, _r__{_v_}, _r__{_ṽ_}, _t_,
_r′__{_v_}, _r′__{_ṽ_}, _t’_ by |_u_|, C_{_u_}, C_{_ū_}, ρ, C′_{_u_},
C′_{_ū_}, ρ′
ρ′ = (-|_u_| C_{_u_} + ρ)/√(1 - _u²_),
C’_{_u_} = (C_{_u_} - |_u_|ρ)/√(1 - _u²_),
C′_{_ū_} = C_{_ū_},
E) (C_{_u_} - |_u_|ρ)/√(1 - _u²_) = σ(E + [_u_M])_{_u_},
C_{_ū_} = σ (E + [_u_M])_{_u_}/√(1 - _u²_).
In consideration of the manner in which σ enters into these relations,
it will be convenient to call the vector C - ρ_u_ with the components
C_{_u_} - ρ|_u_| in the direction of _u_, and C′_{_ū_} in the directions
_ū_ perpendicular to _u_ the “Convection current.” This last vanishes
for σ = 0.
We remark that for ε = 1, μ = 1 the equations _e′_ = E′, _m′_ = M′
immediately lead to the equations _e_ = E, _m_ = M by means of a
reciprocal Lorentz-transformation with -_u_ as vector; and for σ = 0,
the equation C′ = 0 leads to C = ρ_u_; that the fundamental equations of
Äther discussed in § 2 becomes in fact the limitting case of the
equations obtained here with ε = 1, μ = 1, σ = 0.
§ 9. The Fundamental Equations in Lorentz’s Theory.
Let us now see how far the fundamental equations assumed by Lorentz
correspond to the Relativity postulate, as defined in §8. In the article
on Electron-theory (Ency., Math., Wiss., Bd. V. 2, Art 14) Lorentz has
given the fundamental equations for any possible, even magnetised bodies
(see there page 209, Eqn XXX′, formula (14) on page 78 of the same
(part).
(III_a″_) Curl (H - [_u_E]) = J + _d_D/_dt_ + _u_ div D
- curl [_u_D].
(I″) div D = ρ
(IV″) curl E = - _d_B/_dt_, Div B = 0 (V′)
Then for moving non-magnetised bodies, Lorentz puts (page 223, 3) μ = 1,
B = H, and in addition to that takes account of the occurrence of the
di-electric constant ε, and conductivity σ according to equations
(ε_q_XXXIV″, p. 327) D - E = (ε - 1) {E + [_u_B]}
(ε_q_XXXIII′, p. 223) J = σ(E + [_u_B])
Lorentz’s E, D, H are here denoted by E, M, _e_, _m_ while J denotes the
conduction current.
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, = _u_ρ (the current being zero for σ = 0,
(29) Curl [H - (_u_, E)] = C + _d_D/_dt_ - curl [_u_D],
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 - [_u_D] = H -
[_u_D] (M - [_u_E] = _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 - [_u_E]. 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 + [_u_B])_{_u_} = (D - E)_{_u_}
and for a perpendicular direction ū,
(ε - 1) [E + (_u_B)]_{_u_} = (1 - _u²_) (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_{_ū_} are equal to the
components of σ (E + [_u_ B]) multiplied by √(1 - _u²_) or 1 / √(1 -
_u²_) respectively.
§10. Fundamental Equations of E. Cohn.
E. Cohn assumes the following fundamental equations.
(31) Curl (M + [_u_ E]) = _d_E/_dt_ + u div. E + J
- Curl [E - (_u_. M)] = _d_M/_dt_ + u div. M.
(32) J = σ E, = ε E - [_u_ M], M = μ (_m_ + [_u_ E.])
where E M are the electric and magnetic field intensities (forces), E, M
are the electric and magnetic polarisation (induction). The equations
also permit the existence of true magnetism; if we do not take into
account this consideration, div. M. is to be put = 0.
An objection to this system of equations, is that according to these,
for ε = 1, μ = 1, the vectors force and induction do not coincide. If in
the equations, we conceive E and M and not E - (U. M), and M + [U E] as
electric and magnetic forces, and with a glance to this we substitute
for E, M, E, M, div. E, the symbols _e_, M, E + [U M], _m_ - [_u_ _e_],
ρ, then the differential equations transform to our equations, and the
conditions (32) transform into
J = σ(E + [_u_ M])
_e_ + [_u_, (_m_ - [_u_ _e_])] = ε(E + [_u_ M])
M - [_u_, (E + _u_ M)] = μ(_m_ - [_u_ _e_])
then in fact the equations of Cohn become the same as those required by
the relativity principle, if errors of the order _u²_ are neglected in
comparison to 1.
It may be mentioned here that the equations of Hertz become the same as
those of Cohn, if the auxiliary conditions are
(33) E = εE, M = μM, J = σE.
§11. Typical Representations of the Fundamental Equations.
In the statement of the fundamental equations, our leading idea had been
that they should retain a covariance of form, when subjected to a group
of Lorentz-transformations. Now we have to deal with ponderomotive
reactions and energy in the electro-magnetic field. Here from the very
first there can be no doubt that the settlement of this question is in
some way connected with the simplest forms which can be given to the
fundamental equations, satisfying the conditions of covariance. In order
to arrive at such forms, I shall first of all put the fundamental
equations in a typical form which brings out clearly their covariance in
case of a Lorentz-transformation. Here I am using a method of
calculation, which enables us to deal in a simple manner with the
space-time vectors of the 1st, and 2nd kind, and of which the rules, as
far as required are given below.