Part 6
Now it is quite apparent that the system of values
_x₁_ = ω₁, _x₂_ = ω₂, _x₃_ = ω₃, _x₄_ = ω₄
is transformed into the values
_x₁′_ = ω₁′, _x₂′_ = ω₂′, _x₃′_ = ω₃′, _x₄′_ = ω₄′
in virtue of the Lorentz-transformation (10), (11), (12).
The dashed system has got the same meaning for the velocity _u′_ after the transformation as the first system of values has got for _u_ before transformation.
If in particular the vector _v_ of the special Lorentz-transformation be equal to the velocity vector _u_ of matter at the space-time point (_x₁_, _x₂_, _x₃_, _x₄_) then it follows out of (10), (11), (12) that
ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ = _i_
Under these circumstances therefore, the corresponding space-time point has the velocity _v′_ = 0 after the transformation, it is as if we transform to rest. We may call the invariant ρ√(1 - _u²_) the rest-density of Electricity.[16]
§ 5. Space-time Vectors. Of the 1st and 2nd kind.
If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general _relativity theorem_. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.
Let us take a linear homogeneous transformation,
$$ \begin{vmatrix} x_{1} x_{2} x_{3} x_{4} \end{vmatrix} = \begin{vmatrix} a_{1 1} & a_{1 2} & a_{1 3} & a_{1 4} a_{2 1} & a_{2 2} & a_{2 3} & a_{2 4} a_{3 1} & a_{3 2} & a_{3 3} & a_{3 4} a_{4 1} & a_{4 2} & a_{4 3} & a_{4 4} \end{vmatrix} \begin{vmatrix} x_{1}' x_{2}' x_{3}' x_{4}' \end{vmatrix} $$
the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, while _a₄₁_, _a₄₂_, _a₄₃_, are purely imaginary, but _a₄₄_ is real and > 0, and _x₁²_ + _x₂²_ + _x₃²_ + _x₄²_ transforms into _x₁′²_ + _x₂′²_ + _x₃′²_ + _x₄′²_. The operation shall be called a general Lorentz transformation.
(This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski’s notation, _x₁_ = _a₁₁x₁′_ + _a₁₂x₂′_+ _a₁₃x₃′_+ _a₁₄x₄′_.)
If we put _x₁′_ = _x′_, _x₂′_ = _y′_, _x₃′_ = _z′_, _x₄′_ = _it′_, then immediately there occurs a homogeneous linear transformation of (_x_, _y_, _z_, _t_) to (_x′_, _y′_, _z′_, _t′_) with essentially real co-efficients, whereby the aggregate -_x²_ - _y²_ - _z²_ + _t²_ transforms into -_x′²_ - _y′²_ - _z′²_ + _t′²_, and to every such system of values _x_, _y_, _z_, _t_ with a positive _t_, for which this aggregate > 0, there always corresponds a positive _t’_; this last is quite evident from the continuity of the aggregate _x_, _y_, _z_, _t_.
The last vertical column of co-efficients has to fulfil the condition 22) _a₁₄²_ + _a₂₄²_ + _a₃₄²_ + _a₄₄²_ = 1.
If _a₁₄_ = _a₂₄_ = _a₃₄_ = 0, then _a₄₄_ = 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.
If _a₁₄_, _a₂₄_, _a₃₄_ are not all zero, and if we put _a₁₄_ : _a₂₄_ : _a₃₄_ : _a₄₄_ = _v__{_x_} : _v__{_y_} : _v__{_z_} : _i_
_q_ = √(_v__{_x_}² + _v__{_y_}² +_v__{_z_}²) < 1.
On the other hand, with every set of values of _a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_ which in this way fulfil the condition 22) with real values of _v__{_x_}, _v__{_y_}, _v__{_z_}, we can construct the special Lorentz transformation (16) with (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (_a₁₄_, _a₂₄_, _a₃₄_, _a₄₄_) can be supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.
The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where these are the values of (_x₁′_, _x₂′_, _x₃′_, _x₄′_), obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (_x₁_, _x₂_, _x₃_, _x₄_) in the expression (21).
Besides the time-space vector of the 1st kind (_x₁_, _x₂_, _x₃_, _x₄_) we shall also make use of another space-time vector of the first kind (_y₁_, _y₂_, _y₃_, _y₄_), and let us form the linear combination
(23) _f₂₃_(_x₂__y₃_ - _x₃__y₂_) + _f₃₁_(_x₃__y₁_ - _x₁__y₃_) + _f₁₂_(_x₁__y₂_ - _x₂__y₁_) + _f₁₄_(_x₁__y₄_ - _x₄__y₁_) + _f₂₄_(_x₂__y₄_ - _x₄__y₂_) + _f₃₄_(_x₃__y₄_ - _x₄__y₃_)
with six coefficients _f₂₃_--_f₃₄_. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors.
_x₁_, _x₂_, _x₃_; _y₁_, _y₂_, _y₃_; _f₂₃_, _f₃₁_, _f₁₂_; _f₁₄_, _f₂₄_, _f₃₄_ and the constants _x₄_ and _y₄_, at the same time it is symmetrical with regard the indices (1, 2, 3, 4).
If we subject (_x₁_, _x₂_, _x₃_, _x₄_) and (_y₁_, _y₂_, _y₃_, _y₄_) simultaneously to the Lorentz transformation (21), the combination (23) is changed to:
(24) _f₂₃′_(_x₂′__y₃′_ - _x₃′__y₂′_) + _f₃₁_(_x₃′__y₁′_ - _x₁′__y₃′_) + _f₁₂_ (_x₁′__y₂′_ - _x₂′__y₁′_) + _f₁₄′_(_x₁′__y₄′_) - _x₄′__y₁′_) + _f₂₄′_(_x₂′__y₄′_ - _x₄′__y₂′_) + _f₃₄′_(_x₃′__y₄′_ - _x₄′__y₃′_),
where the coefficients _f₂₃′_, _f₃₁′_, _f₁₂′_, _f₁₄′_, _f₂₄′_, _f₃₄′_, depend solely on (_f₂₃_ _f₂₄_) and the coefficients _a₁₁_ ... _a₄₄_.
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 (_pu__{_x_}, _pu__{_y_}, _pu__{_z_}, _i_ρ) (convection-current, and charge density ρ_i_) is transformed as a space time vector of the 1st kind, further (_m__{_x_}, _m__{_y_}, _m__{_z_}, -_ie__{_x_}, -_ie__{_y_}, -_ie__{_z_}) (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 (ρ_u_, _i_ρ) 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,—
(ρ_u_, _i_ρ) 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_(_f₂₃__f₁₄_ + _f₃₁__f₂₄_ + _f₁₂__f₃₄_).
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._ _m__{_x_} = 0, _e__{_x_} = 0. Then
(_m__{_x_}, -_i e__{_x_})² + (_m__{_y_}, -_i e__{_y_})² ≠ 0.
Therefore (_e__{_y_} + _i m__{_y_})/(_e__{_x_} + _i e__{_x_}) is different from +_i_, and we can therefore define a complex argument (φ + _i_ψ) in such a manner that
tan (φ + _i_ψ)
_e__{_y_} + _i m__{_y_} = ------------------------- _e__{_x_} + _i m__{_x_}
If then, by referring back to equations (9), we carry out the transformation (1) through the angle ψ and a subsequent rotation round the Z-axis through the angle φ, we perform a Lorentz-transformation at the end of which _m__{_y_} = 0, _e__{_y_} = 0, and therefore _m_ and _e_ shall both coincide with the new Z-axis. Then by means of the invariants _m²_ - _e²_, (_me_) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.
§ 6. Concept of Time.
By the Lorentz transformation, we are allowed to effect certain _changes_ of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves.[18] In fact, the following considerations will prove conclusive.
Let a reference system (_x_, _y_, _z_, _t_) for space time points (events) be somehow known. Now if a space point A (_x₀_, _y₀_, _z₀_) the time _t₀_ be compared with a space point P (_x_, _y_, _z_) at the time _t_, and if the difference of time _t_ - _t₀_, (let _t_ > _t₀_) be less than the length A P _i.e._ less than the time required for the propagation of light from A to P, and if _q_ = (_t_ - _t₀_)/(A P) < 1, then by a special Lorentz transformation, in which A P is taken as the axis, and which has the moment _q_, we can introduce a time parameter _t′_, which (see equation 11, 12, § 4) has got the same value _t′_ = _0_ for both space-time points (A, _t₀_), and (P, t). So the two events can now be comprehended to be simultaneous.
Further, let us take at the same time _t₀_ = 0, two different space-points A, B, or three space-points (A, B, C) which are not in the same space-line, and compare therewith a space point P, which is outside the line A B, or the plane A B C, at another time _t_, and let the time difference _t_ - _t₀_ (t > _t₀_) be less than the time which light requires for propagation from the line A B, or the plane (A B C) to P. Let q be the quotient of (_t_ - _t₀_) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C is the axis, and q is the moment, then all the three (or four) events (A, _t₀_), (B, _t₀_), (C, _t₀_) and (P, t) are simultaneous.
If four space-points, which do not lie in one plane, are conceived to be at the same time _t₀_, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points.
To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.
## PART II. ELECTRO-MAGNETIC PHENOMENA.
§ 7. Fundamental Equations for bodies at rest.
After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limiting case ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us—when proper fundamental data are given—to obtain the following quantities at every place and time, and therefore at every space-time point as functions of (_x_, _y_, _z_, _t_):—the vector of the electric force E, the magnetic induction M, the electrical induction _e_, the magnetic force _m_, the electrical space-density ρ, the electric current s (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector _v_, the velocity of matter.
The relations in question can be divided into two classes.
Firstly—those equations, which,—when _v_, the velocity of matter is given as a function of (_x_, _y_, _z_, _t_),—lead us to a knowledge of other magnitude as functions of _x_, _y_, _z_, _t_—I shall call this first class of equations the fundamental equations—
Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector _u_ as functions of (_x_, _y_, _z_, _t_).
For the case of bodies at rest, _i.e._ when _u_ (_x_, _y_, _z_, _t_) = 0 the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are;—
(1) The Differential Equations:—which contain no constant referring to matter:—
(_i_) Curl _m_ - δ_e_/δ_t_ = C, (_ii_) div _e_ = lρ. (_iii_) Curl E + δM/δ_t_ = 0, (_iv_) Div M = 0.
(2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves _i.e._ for isotopic bodies;—they are comprised in the equations
(V) _e_ = ε E, M = μ_m_, C = σE.
where ε = dielectric constant, μ = magnetic permeability, σ = the conductivity of matter, all given as function of _x_, _y_, _z_, _t_; _s_ is here the conduction current.
By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,
_x₁_ = _x_, _x₂_ = _y_, _x₃_ = _z_, _x₄_ = _it_,
and write _s₁_, _s₂_, _s₃_, _s₄_ for C_{_x_}, C_{_y_}, C_{_z_} (√-1)ρ.
Further _f₂₃_, _f₃₁_, _f₁₂_, _f₁₄_, _f₂₄_, _f₃₄_
for _m__{_x_}, _m__{_y_}, _m__{_z_}, -_i_(_e__{_x_}, _e__{_y_}, _e__{_z_}),
and F₂₃, F₃₁, F₁₂, F₁₄, F₂₄, F₃₄
for M_{_x_}, M_{_y_}, M_{_z_}, -_i_(E_{_x_}, E_{_y_}, E_{_z_})
lastly we shall have the relation _f__{k h} = - _f__{_h k_}, _F__{_k h_} = -_F__{_h k_}, (the letter _f_, F shall denote the field, _s_ the (_i.e._ current).
Then the fundamental Equations can be written as
(A) ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = s₁
∂_f₂₁_/∂_x₁_ + + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = s₂
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + + ∂_f₃₄_/∂_x₄_ = s₃
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = s₄
and the equations (3) and (4), are
∂F₃₄/∂_x₂_ + ∂F₄₂/∂_x₃_ + ∂F₂₃/∂_x₄_ = 0
∂F₄₃/∂_x₁_ + + ∂F₁₄/∂_x₃_ + ∂F₃₁∂_x₄_ = 0
∂F₂₄/∂_x₁_ + ∂F₄₁/∂_x₂_ + + ∂F₁₂/∂_x₄_ = 0
∂F₃₂/∂_x₁_ + ∂F₁₃/∂_x₂_ + ∂F₂₁/∂_x₃_ = 0
§ 8. The Fundamental Equations.
We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.
The first Axion shall be,—
When a detached region[19] of matter is at rest at any moment, therefore the vector _u_ is zero, for a system (_x_, _y_, _z_, _t_)—the neighbourhood may be supposed to be in motion in any possible manner, then for the space-time point _x_, _y_, _z_, _t_, the same relations (A) (B) (V) which hold in the case when all matter is at rest, shall also hold between ρ, the vectors C, _e_, _m_, _M_, _E_ and their differentials with respect to _x_, _y_, _z_, _t_. The second axiom shall be:—
Every velocity of matter is < 1, smaller than the velocity of propagation of light.[20]
The fundamental equations are of such a kind that when (_x_, _y_, _z_, _it_) are subjected to a Lorentz transformation and thereby (_m_ - _ie_) and (_M_ - _iE_) are transformed into space-time vectors of the second kind, (C, _i_ρ) as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.
Shortly I can signify the third axiom as:—
(_m_, -_ie_), and (_M_, -_iE_) are space-time vectors of the second kind, (C, _i_p) is a space-time vector of the first kind.
This axiom I call the Principle of Relativity.
In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.
According to the second axiom, the magnitude of the velocity vector | _u_ | is < 1 at any space-time point. In consequence, we can always write, instead of the vector _u_, the following set of four allied quantities
ω₁ = u_{_x_}/√(1 - _u²_), ω₂ = u_{_y_}/√(1 - u²), ω₃ = u_{_z_}/√(1 - u²), ω₄ = _i_/√(1 - u²)
with the relation
(27) ω₁² + ω₂² + ω₃² + ω₄² = - |
From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space-time vector of the 1st kind.
Let us now fix our attention on a certain point (_x_, _y_, _z_) of matter at a certain time (_t_). If at this space-time point _u_ = 0, then we have at once for this point the equations (_A_), (_B_) (_V_) of § 7. If _u_ ≠ 0, then there exists according to 16), in case | _u_ | < 1, a special Lorentz-transformation, whose vector _v_ is equal to this vector _u_ (_x_, _y_, _z_, _t_), and we pass on to a new system of reference (_x′_ _y′_ _z′_ _t′_) in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values 28) ω′₁ = 0, ω′₂ = 0, ω′₃ = 0, ω′₄ = _i_, therefore the new velocity vector ω′ = 0, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point (_x′_ _y′_ _z′_ _t_) involves the newly introduced magnitude (_u′_ ρ′, C′, _e′_, _m′_, _E′_, _M′_) and their differential quotients with respect to (_x′_, _y′_, _z′_, _t′_) in the same manner as the original equations for the point (_x_, _y_, _z_, _t_). But according to the first axiom, when _u′_ = 0, these equations must be exactly equivalent to
(1) the differential equations (_A′_), (_B′_), which are obtained from the equations (_A_), (_B_) by simply dashing the symbols in (_A_) and (_B_).
(2) and the equations
(V′) _e′_ = ε_E′_, _M’_ = μ_m′_, _C′_ = σ_E′_
where ε, μ, σ are the dielectric constant, magnetic permeability, and conductivity for the system (_x′_ _y′_ _z′_ _t′_) _i.e._ in the space-time point (_x_ _y_, _z_ _t_) of matter.
Now let us return, by means of the reciprocal Lorentz-transformation to the original variables (_x_, _y_, _z_, _t_), and the magnitudes (_u_, ρ, C, _e_, _m_, _E_, _M_) and the equations, which we then obtain from the last mentioned, will be the fundamental equations sought by us for the moving bodies.
Now from § 4, and § 6, it is to be seen that the equations _A_), as well as the equations _B_) are covariant for a Lorentz-transformation, _i.e._ the equations, which we obtain backwards from _A′_) _B′_), must be exactly of the same form as the equations _A_) and _B_), as we take them for bodies at rest. We have therefore as the first result:—
The differential equations expressing the fundamental equations of electrodynamics for moving bodies, when written in ρ and the vectors C, _e_, _m_, E, M, are exactly of the same form as the equations for moving bodies. The velocity of matter does not enter in these equations. In the vectorial way of writing, we have
I) curl _m_ - ∂_e_/∂_t_ = C₁,
II) div _e_ = ρ
III) curl E + ∂M/∂_t_ = 0
IV) div M = 0
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′)