Chapter V: 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′)

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.

A system of magnitudes _a__{_h_ _k_} formed into the matrix

| _a₁₁_...................._a__{1 _q_} |
| |
| |
| |
| _a__{_p_ 1}..........._a__{_p_ _q_} |

arranged in _p_ horizontal rows, and _q_ vertical columns is called a _p_ × _q_ series-matrix, and will be denoted by the letter A.

If all the quantities _a__{_h_ _k_} are multiplied by C, the resulting matrix will be denoted by CA.

If the roles of the horizontal rows and vertical columns be intercharged, we obtain a _q_ × _p_ series matrix, which will be known as the transposed matrix of A, and will be denoted by Ā.

Ā = | _a₁₁_ ...................... _a__{_p_ 1} |
| |
| _a__{1 _q_} ............ _a__{_p_ _q_} |

If we have a second _p_ × _q_ series matrix B,

B = | _b₁₁_ ......................... _b₁__{_q_} |
| |
| _b__{_p_ 1} ............. b_{_p_ _q_} |

then A + B shall denote the _p_ × _q_ series matrix whose members are _a__{_h_ _k_} + _b__{_h_ _k_}.

2⁰ If we have two matrices

A = | _a₁₁_ ..................... _a__{1 _q_} |
| |
| _a__{_p_ 1} ........... _a__{_p_ _q_} |

B = | _b__{1 1} .............. _b__{1 _r_} |
| |
| _b__{_q_ 1} .......... _b__{_p_ _r_} |

where the number of horizontal rows of B, is equal to the number of vertical columns of A, then by AB, the product of the matrices A and B, will be denoted the matrix

C = | _c₁₁_ ...................... _c__{1 _r_} |
| |
| _c__{_p_ _r_} ........... _c__{_p_ _p_} |

where _c__{_h_ _k_} = _a__{_h_ 1} _b₁__{_k_} + _a__{_h_ 2} _b__{2 _h_} + ... _a__{_k_ _s_} _b__{_s_ _k_} + ... + _a__{_k_ _q_} _b__{_q_ _h_}

these elements being formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law (AB)S = A(BS) holds, where S is a third matrix which has got as many horizontal rows as B (or AB) has got vertical columns.

For the transposed matrix of C = BA, we have Ċ = ḂĀ

3⁰. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows.

As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 × 4 series) with the elements.

(34) | e₁₁ e₁₂ e₁₃ e₁₄ | = | 1 0 0 0 |
| e₂₁ e₂₂ e₂₃ e₂₄ | | 0 1 0 0 |
| e₃₁ e₃₂ e₃₃ e₃₄ | | 0 0 1 0 |
| e₄₁ e₄₂ e₄₃ e₄₄ | | 0 0 0 1 |

For a 4 × 4 series-matrix, Det A shall denote the determinant formed of the 4 × 4 elements of the matrix. If det A ≠ 0, then corresponding to A there is a reciprocal matrix, which we may denote by A⁻¹ so that A⁻¹A = 1.

A matrix

_f_ = | 0 _f₁₂_ _f_₁₃ _f₁₄_ |
| _f_₂₁ 0 _f₂₃_ _f₂₄_ |
| _f₃₁_ _f_₃₂ 0 _f₃₄_ |
| _f_₄₁ _f_₄₂ _f_₄₃ 0 |

in which the elements fulfil the relation _f__{_h_ _k_} = -_f__{_h_ _k_}, is called an alternating matrix. These relations say that the transposed matrix _ḟ_ = -_f_. Then by _f_^{*} will be the _dual_, alternating matrix

(35)

_f_^{*} = | 0 _f₃₄_ _f_₄₂ _f₂₃_ |
| _f_₄₃ 0 _f₁₄_ _f₃₁_ |
| _f₂₄_ _f_₄₁ 0 _f₁₂_ |
| _f_₃₂ _f_₁₃ _f_₂₁ 0 |

Then (36) _f_* _f_ = _f₃₄_ _f₂₂_ + _f₄₂_ _f₃₁_ + _f₃₂_ _f₂₄_

_i.e._ We shall have a 4 × 4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).

The determinant of _f_ is therefore the square of the combination, by Det^{½}_f_ we shall denote the expression

Det^{½}_f_
= _f₃₂_ _f₁₄_ _f₁₃_ _f₂₄_ + _f₂₁_ _f₃₄_·

4⁰. A linear transformation

_x__{_h_} = α_{_h_1} _x₁′_ + α_{_h_2} _x₂_′ + α_{_h_3} _x₃′_ + α_{_h_4} _x₄′_ (_h_ = 1,2,3,

which is accomplished by the matrix

A = | α₁₁, α₁₂, α₁₃, α₁₄ |
| |
| α₂₁, α₂₂, α₂₃, α₂₄ |
| |
| α₃₁, α₃₂, α₃₃, α₃₄ |
| |
| α₄₁, α₄₂, α₄₃, α₄₄ |

will be denoted as the transformation A.

By the transformation A, the expression

_x²₁_ + _x²₂_ + _x²₃_ + _x²₄_ is changed into the quadratic for _m_ ∑ α_{_hk_} _x__{_h_}′ _x__{_k_}′,

where α_{_hk_} = α_{1_k_} α_{1_k_} + α_{2_h_} α_{2_k_} + α_{3_h_} α_{3_k_} + α_{4_h_} α_{4_k_} are the members of a 4 × 4 series matrix which is the product of Ā A, the transposed matrix of A into A. If by the transformation, the expression is changed to

_x′₁²_ + _x₂′_^2 + _x₃′_^2 + _x′₄²_,

we must have Ā A = 1.

A has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of A) it follows out of (39) that (Det A)² = 1, or Det A = ± 1.

From the condition (39) we obtain

A⁻¹ = Ā,

_i.e._ the reciprocal matrix of A is equivalent to the transposed matrix of A.

For A as Lorentz transformation, we have further Det A = +1, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and _a₄₄_ > 0.

5⁰. A space time vector of the first kind[21] which s represented by the 1 × 4 series matrix,

(41) _s_ = |_s₁_ _s₂_ _s₃_ _s₄_|

is to be replaced by _s_A in case of a Lorentz transformation

A. _i.e._ _s′_ = | _s₁′_ _s₂′_ _s₃′_ _s₄′_| = |_s₁_ _s₂_ _s₃_ _s₄_|
A;

A space-time vector of the 2nd kind[22] with components _f₂₃_ ... _f₃₄_ shall be represented by the alternating matrix

(42) _f_ = | 0 _f_₁₂ _f₁₃_ _f₁₄_ |

|_f₂₁_ 0 _f_₂₃ _f₂₄_ |

|_f_₃₁ _f₃₂_ 0 _f₃₄_ |

|_f_₄₁ _f_₄₂ _f_₄₃ 0 |

and is to be replaced by A⁻¹ _f_ A in case of a Lorentz transformation [see the rules in § 5 (23) (24)]. Therefore referring to the expression (37), we have the identity Det^{½} (Ā _f_ A) = Det A. Det^{½} _f_. Therefore Det^{½} _f_ becomes an invariant in the case of a Lorentz transformation [see eq. (26) See. § 5].

Looking back to (36), we have for the dual matrix (Ā_f_*A) (A⁻¹_f_A) = A⁻¹_f_*_f_A = Det^{½} function. A⁻¹A = Det^{½}_f_ from which it is to be seen that the dual matrix _f_* behaves exactly like the primary matrix _f_, and is therefore a space time vector of the II kind; _f_* is therefore known as the dual space-time vector of _f_ with components (_f₁₄_, _f₂₄_, _f₃₄_,), (_f₂₃_}, _f₃₁_, _f₁₂_).

6. If _w_ and _s_ are two space-time rectors of the 1st kind then by _w_ _ṡ_ (as well as by _s_ _ẇ_) will be understood the combination (43) _w₁_ _s₁_ + _w₂_ _s₂_ + _w₃_ _s₃_ + _w₄_ _s₄_.

In case of a Lorentz transformation A, since (_w_A) (Ā_ṡ_) = _w_ _s_, this expression is invariant.—If _w_ _ṡ_ = 0, then _w_ and _s_ are perpendicular to each other.

Two space-time rectors of the first kind (_w_, _s_) gives us a 2 × 4 series matrix

| _w₁_ _w₂_ _w₃_ _w₄_ |
| _s₁_ _s₂_ _s₃_ _s₄_ |

Then it follows immediately that the system of six magnitudes (44)

_w₂_ _s₃_ - _w₃_ _s₂_,
_w₃_ _s₁_ - _w₁_ _s₃_,
_w₁_ _s₂_ - _w₂_ _s₁_,
_w₁_ _s₄_ - _w₄_ _s₁_,
_w₂_ _s₄_ - _w₄_ _s₂_,
_w₃_ _s₄_ - _w₄_ _s₃_,

behaves in case of a Lorentz-transformation as a space-time vector of the II kind. The vector of the second kind with the components (44) are denoted by [_w_, _s_]. We see easily that Det^{½} [_w_, _s_] = 0. The dual vector of [_w_, _s_] shall be written as [_w_, _s_].

If _ẇ_ is a space-time vector of the 1st kind, _f_ of the second kind, _w_ _f_ signifies a 1 × 4 series matrix. In case of a Lorentz-transformation A, _w_ is changed into _w′_ = _w_A, _f_ into _f′_ = A⁻¹ _f_ A,—therefore _w′_ _f′_ becomes = (_w_A A⁻¹ _f_ A) = _w_ _f_ A _i.e._ _w_ _f_ is transformed as a space-time vector of the 1st kind.[23] We can verify, when _w_ is a space-time vector of the 1st kind, _f_ of the 2nd kind, the important identity

(45) [_w_, _w__f_] + [_w_, _w__f_*]* = (_w_] _ẇ_)_f_.

The sum of the two space time vectors of the second kind on the left side is to be understood in the sense of the addition of two alternating matrices.

For example, for ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_,

ω_f_ = | _i__f_₄₁, _i__f_₄₂, _i__f_₄₃, 0 |;
ω_f_* = | _i__f_₃₂, _i__f_₁₃, _i__f_₂₁, 0 |

[ω · ω_f_] = 0, 0, 0, _f_₄₁, _f_₄₂, _f_₄₃;
[ω · ω_f_*]* = 0, 0, 0, _f_₃₂, _f_₁₃, _f_₂₁.

The fact that in this special case, the relation is satisfied, suffices to establish the theorem (45) generally, for this relation has a covariant character in case of a Lorentz transformation, and is homogeneous in (ω₁, ω₂, ω₃, ω₄).

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants ε μ, σ will be introduced.

Instead of the space vector _u_, the velocity of matter, we shall introduce the space-time vector of the first kind ω with the components.

ω₁ = _u__{_x_}/√(1 - _u²_),
ω₂ = _u__{_y_}/√(1 - _u²_),
ω₃ = _u__{_z_}/√(1 - _u²_),
ω₄ = _i_/√(1 - _u²_).

(40) where ω₁² + ω₂² + ω₃² + ω₄² = -1 and -_i_ω₄ > 0.

By F and _f_ shall be understood the space time vectors of the second kind M - _i_E, _m_ - _ie_.

In Φ = ωF, we have a space time vector of the first kind with components

Φ₁ = ω₂F₁₂ + ω₃F₁₃ + ω₄F₁₄

Φ₂ = ω₁F₂₁ + ω₃F₂₃ + ω₄F₂₄

Φ₃ = ω₁F₃₁ + ω₂F₃₂ + ω₄F₃₄

Φ₄ = ω₁F₄₁ + ω₂F₄₂ + ω₃F₄₃

The first three quantities (φ₁, φ₂, φ₃) are the components of the space-vector (E + [_u_, M])/√(1 - _u²_),

and further (φ₄ = _i_[_u_ E]/√(1 - _u²_).

Because F is an alternating matrix,

(49) ωΦ = ω₁ φ₁ + ω₂ Φ₂ + ω₃ Φ₃ + ω₄ Φ₄ = 0.

_i.e._ Φ is perpendicular to the vector ω; we can also write Φ₄ = _i_[ω_{x} Φ₁ + ω_{y} Φ₂ + ω_{z} Φ₃].

I shall call the space-time vector Φ of the first kind as the _Electric Rest Force_.[24]

Relations analogous to those holding between -ωF, E, M, U, hold amongst -ω_f_, _e_, _m_, _u_, and in particular -ω_f_ is normal to ω. The relation (C) can be written as

{C} ω_f_ = εωF.

The expression (ω_f_) gives four components, but the fourth can be derived from the first three.

Let us now form the time-space vector 1st kind, ψ - _i_ω_f_*, whose components are

ψ₁ = -_i_(ω₂ _f₃₄_ + ω₃ _f_₄₂ + ω₄ _f₂₃_)
ψ₂ = -_i_(ω₁ _f_₄₃ + ω₃ _f_₄₄ + ω₄ _f₃₁_)
ψ₃ = -_i_(ω₁ _f₂₄_ + ω₂ _f_₄₁ + ω₄ _f₁₂_)
ψ₄ = -_i_(ω₁ _f_₃₂ + ω₂ _f_₁₃ + ω₃ _f_₂₁)

Of these, the first three ψ₁, ψ₂, ψ₃, are the _x_, _y_, _z_ components of the space-vector 51) (m - (_ue_))/√(1 - _u²_) and further (52) ψ₄ = _i_(_u_m)/√(1 - _u²_).

Among these there is the relation

(53) ωψ = ω₁ ψ₁ + ω₂ ψ₂ + ω₃ ψ₃ + ω₄ ψ₄ = 0

which can also be written as ψ₄ = _i_ (_u__{_x_} ψ₁ + _u__{_y_} ψ₂ + _u__{_z_} ψ₃).

The vector ψ is perpendicular to ω; we can call it the _Magnetic rest-force_.

Relations analogous to these hold among the quantities ωF*, M, E, _u_ and Relation (D) can be replaced by the formula

{ D } -ωF* = μψ_f_*.

We can use the relations (C) and (D) to calculate F and _f_ from Φ and ψ we have

ωF = -Φ, ωF* = -_i_μψ, ω_f_ = -εΦ, ω_f_* = -_i_ψ.

and applying the relation (45) and (46), we have

F = [ω. Φ] + _i_μ[ω. ψ]* 55)
_f_ = ε[ω. Φ] + _i_[ω. ψ]* 56)

_i.e._

F₁₂ = (ω₁ Φ₁ - ω₂ Φ₁) + _i_μ [ω₃ Ψ₄ - ω₄ ψ₃], etc.
_f₁₂_ = ε(ω₁ Φ₂ - ω₂ φ₁) + _i_ [ω₃ ψ₄ - ω₄ ψ₃]., etc.

Let us now consider the space-time vector of the second kind [Φ ψ], with the components

[ Φ₂ ψ₃ - Φ₃ ψ₂, Φ₃ ψ₁ - Φ₁ ψ₃, Φ₁ ψ₂ - Φ₂ ψ₁ ]
[ Φ₁ ψ₄ - Φ₄ ψ₁, Φ₂ ψ₄ - Φ₄ ψ₂, Φ₃ ψ₄ - Φ₄ ψ₃ ]

Then the corresponding space-time vector of the first kind ω[Φ, ψ] vanishes identically owing to equations 9) and 53)

for ω[Φ.ψ] = -(ωψ)Φ + (ωΦ)ψ

Let us now take the vector of the 1st kind

(57) Ω = _i_ω[Φψ]*

with the components

Ω₁ = -_i_ | ω₂ ω₃ ω₄ |
| Φ₂ Φ₃ Φ₄ |
| ψ₂ ψ₃ ψ₄ |, etc.

Then by applying rule (45), we have

(58) [Φ.ψ] = _i_[ωΩ]*

_i.e._ Φ₁ψ₂ - Φ₂ψ₁ = _i_(ω₃Ω₄ - ω₄Ω₃) etc.

The vector Ω fulfils the relation

(ωΩ) = ω₁Ω₁ + ω₂Ω₂ + ω₃Ω₃ + ω₄Ω₄ = 0,

(which we can write as Ω₄ = _i_(ω_{x}Ω₁ + ω_{y}Ω₂ + ω_{z}Ω₃) and Ω is also normal to ω. In case ω = 0, we have Φ₄ = 0, ψ₄ = 0, Ω₄ = 0, and

[Ω₁, Ω₂, Ω₃ = | Φ₁ Φ₂ Φ₃ |
|ψ₁ ψ₂ ψ₃ |.

I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.

As for the relation E), which introduces the conductivity σ we have -ωS = -(ω₁_s₁_ + ω₂_s₂_ + ω₃_s₃_ + ω₄_s₄_) = (- | _u_ | C_{_u_} + ρ)/√(1 - _u²_) = ρ′.

This expression gives us the rest-density of electricity (see §8 and §4).

Then 61) = _s_ + (ω_ṡ_)ω represents a space-time vector of the 1st kind, which since ωω = -1, is normal to ω, and which I may call the rest-current. Let us now conceive of the first three component of this vector as the (_x_-_y_-_z_) co-ordinates of the space-vector, then the component in the direction of _u_ is

C_{_u_} - (| _u_ | ρ′)/√(1 - _u²_)
= (_c__{_u_} - | _u_ |ρ)/√(1 - _u²_)
= J_{_u_}/(1 - _u²_)

and the component in a perpendicular direction is C_{_u_} = J_{_ū_}.

This space-vector is connected with the space-vector J = C - ρ_u_, which we denoted in §8 as the conduction-current.

Now by comparing with Φ = -ωF, the relation (E) can be brought into the form

{E} _s_ + (ω_ṡ_)ω = - σωF,

This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to ω.

Lastly, we shall transform the differential equations (A) and (B) into a typical form.

§12. The Differential Operator Lor.

A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | S_{_kh_} |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |

with the condition that in case of a Lorentz transformation it is to be replaced by ĀSA, may be called a space-time matrix of the II kind. We have examples of this in:—

1) the alternating matrix _f_, which corresponds to the space-time vector of the II kind,—

2) the product _f_F of two such matrices, for by a transformation A, it is replaced by (A⁻¹_f_A·A⁻¹FA) = A⁻¹_f_FA,

3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time vectors of the 1st kind, the 4 × 4 matrix with the element S_{_hk_} = ω_{_h_}Ω_{_k_},

lastly in a multiple L of the unit matrix of 4 × 4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.

We shall have to do constantly with functions of the space-time point (_x_, _y_, _z_, _it_), and we may with advantage

employ the 1 × 4 series matrix, formed of differential symbols,—

| ∂/∂_x_, ∂/∂_y_, ∂/∂_z_, ∂/_i_∂_t_,|
or (63) | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ |

For this matrix I shall use the shortened from “lor.”[25]

Then if S is, as in (62), a space-time matrix of the II kind, by lor S′ will be understood the 1 × 4 series matrix

| K₁ K₂ K₃ K₄ |

where K_{_k_} = ∂S_{1_k_}/∂_x₁_ + ∂S_{2_k_}/∂_x₂_ + ∂S_{3_k_}/∂_x₃_ + ∂S_{4_h_}/∂_x₄_.

When by a Lorentz transformation A, a new reference system (_x′₁_ _x′₂_ _x′₃_ _x₄_) is introduced, we can use the operator

lor′ = | ∂/∂_x₁′_ ∂/∂_x₂′_ ∂/∂_x₃′_ ∂/∂_x₄′_ |

Then S is transformed to S′= Ā S A = | S′_{_hk_} |, so by lor 'S′ is meant the 1 × 4 series matrix, whose element are

K’_{_k_} = ∂S′_{1_k_}/∂_x₁′_ + ∂S′_{2_k_}/∂_x₂′_
+ ∂S′_{3_k_}/∂_x₃′_ + ∂S′_{4_k_}/∂_x₄′_.

Now for the differentiation of any function of (_x_ _y_ _z_ _t_) we have the rule ∂/∂_x__{_k_}′ = ∂/∂_x₁_ ∂_x₁_/∂_x__{_k_}′ + ∂/∂_x₂_ ∂_x₂_/∂_x__{_k_}′ + ∂/∂_x₃_ ∂_x₃_/∂_x__{_k_}′ + ∂/∂_x₄_ ∂_x₄_/∂_x__{_k_}′ = ∂/∂_x₁_ _a__{1_k_} + ∂/∂_x₂_ _a__{2_k_} + ∂/∂_x₃_ _a__{3_k_} + ∂/∂_x₄_ _a__{4_k_}.

so that, we have symbolically lor′ = lor A.

Therefore it follows that

lor ′S′ = lor (A A⁻¹ SA) = (lor S)A.

_i.e._, lor S behaves like a space-time vector of the first kind.

If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements

| ∂L/∂_x₁_ ∂L/∂_x₂_ ∂L/∂_x₃_ ∂L/∂_x₄_ |

If _s_ is a space-time vector of the 1st kind, then

lor _ṡ_ = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_.

In case of a Lorentz transformation A, we have

lor ′_ṡ′_ = lor A. Ā_s_ = lor _s_.

_i.e._, lor _s_ is an invariant in a Lorentz-transformation.

In all these operations the operator lor plays the part of a space-time vector of the first kind.

If _f_ represents a space-time vector of the second kind,—lor _f_ denotes a space-time vector of the first kind with the components

∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_,
∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_,
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_,
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_

So the system of differential equations (A) can be expressed in the concise form

{A} lor f = -_s_,

and the system (B) can be expressed in the form

{B} log F* = 0.

Referring back to the definition (67) for log _ṡ_, we find that the combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish identically, when _f_ and F* are alternating matrices. Accordingly it follows out of {A}, that

(68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) =
0,

while the relation

(69) lor (lor F*) = 0,

signifies that of the four equations in {B}, only three represent independent conditions.

I shall now collect the results.

Let ω denote the space-time vector of the first kind

(_u_/√(1 - _u²_}), _i_/√(1 - _u²_))

(_u_ = velocity of matter),

F the space-time vector of the second kind (M,-_i_E)

(M = magnetic induction, E = Electric force,

_f_ the space-time vector of the second kind (_m_,-_ie_)

(_m_ = magnetic force, _e_ = Electric Induction.

_s_ the space-time vector of the first kind (C, _i_ρ)

(ρ = electrical space-density, C - ρ_u_ = conductivity current,

ε = dielectric constant, μ = magnetic permeability,

σ = conductivity,

then the fundamental equations for electromagnetic processes in moving bodies are[26]

{A} lor _f_ = -_s_

{B} log F* = 0

{C} ω_f_ = εωF

{D} ωF* = μω_f_*

{E} _s_ + (ω_ṡ_), _w_ = - σωF.

ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time vectors of the first kind are all normal to ω, and for the system {B}, we have

lor (lor F*) = 0.

Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the motion of matter as well as the vector _u_ as a function of _x_, _y_, _z_, _t_, when proper fundamental data are given.

§ 13. The Product of the Field-vectors _f_ F.

Finally let us enquire about the laws which lead to the determination of the vector ω as a function of (_x_, _y_, _z_, _t_.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

_f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ |
| _f₂₁_ 0 _f₂₃_ _f₂₄_ |
| _f₃₁_ _f₃₂_ 0 _f₃₄_ |
| _f₄₁_ _f₄₂_ _f₄₃_ 0 |

F = | 0 F₁₂ F₁₃ F₁₄ |
| F₂₁ 0 F₂₃ F₂₄ |
| F₃₁ F₃₂ 0 F₃₄ |
| F₄₁ F₄₂ F₄₃ 0 |

are of much importance. Let us write,

(70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |

| S₂₁ S₂₂ - L S₂₃ S₂₄ |

| S₃₁ S₃₂ S₃₃ - L S₃₄ |

| S₄₁ S₄₂ S₄₃ S₄₄ - L |

Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

(72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄
+ _f₂₄_ F₂₄ + _f₃₄_ F₃₄)

Then we shall have

(73) S₁₁ = ½(_f₂₃_ F₂₃ + _f₃₄_ F₃₄ + _f₄₂_ F₄₂ - _f₁₂_ F₁₂
- _f₁₃_ F₁₃ _f₁₄_ F₁₄)

S₁₂ = _f₁₃_ F₃₂ + _f₁₄_ F₄₂ etc....

In order to express in a real form, we write

(74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |

| S₂₁ S₂₂ S₂₃ S₂₄ |

| S₃₁ S₃₂ S₃₃ S₃₄ |

| S₄₁ S₄₂ S₄₃ S₄₄ |

= | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |

| X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |

| X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |

| -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_} |

Now X_{_x_} = ½[_m__{_x_}M_{_x_} - _m__{_y_}M_{_y_} - _m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} - _e__{_z_}E_{_z_}]

so

(75) X_{_y_} = _m__{_x_}M_{_y_} + _e__{_y_}E_{_x_}, Y_{_x_} =
_m__{_y_}M_{_x_} + _e__{_x_}E_{_y_} etc.

X_{_t_} = _e__{_y_}M_{_z_} - _e__{_z_}M_{_y_}, T_{_x_} =
_m__{_x_}E_{_y_} - _m__{_y_}E_{_z_}, etc.

T_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} +
_m__{_z_}M_{_z_} + _e__{_x_}E_{_x_} + _e__{_y_}E_{_y_} +
_e__{_z_}E_{_z_}]

L_{_t_} = ½[_m__{_x_}M_{_x_} + _m__{_y_}M_{_y_} +
_m__{_z_}M_{_z_} - _e__{_x_}E_{_x_} - _e__{_y_}E_{_y_} -
_e__{_z_}E_{_z_}]

These quantities[27] are all real. In the theory for bodies at rest, the combinations (X_{_x_}, X_{_y_}, X_{_z_}, Y_{_z_}, Y_{_y_}, Y_{_z_}, Z_{_x_}, Z_{_y_}, Z_{_z_}) are known as “Maxwell’s Stresses,” T_{_x_}, T_{_y_}, T_{_z_} are known as the Poynting’s Vector, T_{_t_} as the electromagnetic energy-density, and L as the Langrangian function.

On the other hand, by multiplying the alternating matrices of _f_* and F*, we obtain

(77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |

| -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |

| -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |

| -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |

and hence, we can put

(78) _f_F = S - L, F*_f_* = -S - L,

where by L, we mean L-times the unit matrix, _i.e._ the matrix with elements

| L_e__{_hk_} |, (_e__{_hh_} = 1, _e__{_hk_} = 0, _h_ ≠ _k_ _h_, _k_
= 1, 2, 3, 4).

Since here SL = LS, we deduce that,

F*_f_*_f_F = (-S - L)(S - L) = -SS + L²,

and find, since _f_*_f_ = Det^{½}_f_, F*F = Det^{½}F, we arrive at the interesting

conclusion

(79) SS = L² - Det^{½}_f_ Det^{½}F

_i.e._ the product of the matrix S into itself can be expressed as the multiple of a unit matrix—a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations

(80) S_{_h_1} S_{1_k_} + S_{_h_2} S_{2_k_} + S_{_h_3} S_{3_k_} +
S_{_h_4} S_{4_k_} = 0,

_h_, _k_ being unequal indices in the series 1, 2, 3, 4, and

(81) S_{_h_1} S_{1_h_} + S_{_h_2} S_{2_h_} + S_{_h_3} S_{3_h_} +
S{_h_4} S_{4_h_} = L² -
Det^{½}_f_ Det^{½}F,

for _h_ = 1, 2, 3, 4.

Now if instead of F, and _f_ in the combinations (72) and (73), we introduce the electrical rest-force Φ, the magnetic rest-force ψ, and the rest-ray Ω [(55), (56) and (57)], we can pass over to the expressions,—

(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],

(83) S_{_hk_} = - ½ ε Φ [=Φ] _e__{_hk_} - ½ μ ψ [=ψ] _e__{_hk_}
+ ε (Φ_{_h_} Φ_{_k_} - Φ ([=Φ]) ω_{_h_} Ω_{_k_}
+ μ (ψ_{_h_} ψ_{_k_} - Ψ [=ψ] Ω{_h_} ω_{_k_}) - ω_{_h_} ω_{_k_} - εμ
ω_{_h_} Ω_{_k_}
(_h₁_ _k_ = 1, 2, 3, 4).

Here we have

Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²

_e__{_hh_} = 1, _e__{_hk_} = 0 (_h_ ≠ _k_).

The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the right side of (83) as well as S_{_k_ _h_} represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = _i_. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and _e_ = εE, M = μ_m_ on the other hand.

The expression on the right-hand side of (81), which equals

[½ (_m_ M - _e_E)²] + (_em_) (EM),

is >= 0, because (_em_ = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back to 79), we can denote the positive square root of this expression as Det^{1/4} S.

Since _ḟ_ = -_f_, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of S, the following relations from (78),

(84) F_f_ = Ṡ - L, _f_* F* = -Ṡ - L,

Then is

Ṡ - S = | S_{_h_ _k_} - S_{_t_ _k_} |

an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,

(85) S - Ṡ = - (εμ - 1) [ω, Ω],

from which we deduce that [see (57), (58)].

(86) ω (S - Ṡ)* = 0,

(87) ω (S - Ṡ) = (εμ - 1) Ω

When the matter is at rest at a space-time point, ω = 0, then the equation 86) denotes the existence of the following equations

Z_{_y_} = Y_{_z_}, X_{_z_} = Z_{_x_}, Y_{_x_} = X_{_y_},

and from 83),

T_{_x_} = Ω₁, T_{_y_} = Ω₂, T_{_z_} = Ω₃

X_{_t_} = εμΩ₁, Y_{_t_} = εμΩ₂, Z_{_t_} = εμΩ₃

Now by means of a rotation of the space co-ordinate system round the null-point, we can make,

Z_{_y_} = Y_{_z_} = 0, X_{_z_} = Z_{_x_} = 0, X_{_x_} = X_{_y_} = 0,

According to 71), we have

(88) X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} = 0,

and according to 83), T_{_t_} > 0. In special cases, where ω vanishes it follows from 81) that

X_{_x_}² = Y_{_y_}² = Z_{_z_}² = T_{_t_}², = (Det^{1/4} S)²,

and if T, and one of the three magnitudes X_{_x_}, Y_{_y_}, Z_{_z_} are = ±Det^{1/4} S, the two others = -Det^{1/4} S. If Ω does not vanish let Ω ≠ 0, then we have in particular from 80)

T_{_z_} X_{_t_} = 0, T_{_z_} Y_{_t_} = 0, Z_{_z_} T_{_z_} + T_{_z_}
T_{_t_} = 0,

and if Ω₁ = 0, Ω₂ = 0, Z_{_z_} = -T_{_t_} It follows from (81), (see also 83) that

X_{_x_} = -Y_{_y_} = ±Det^{1/4} S,

and -Z_{_z_} = T_{_t_} = √(Det^{½} S + εμΩ₃²) > Det^{1/4}S.

The space-time vector of the first kind

(89) K = lor S,

is of very great importance for which we now want to demonstrate a very important transformation

According to 78), S = L + _f_F, and it follows that

lor S = lor L + lor _f_F.

The symbol ‘lor’ denotes a differential process which in lor _f_F, operates on the one hand upon the components of _f_, on the other hand also upon the components of F. Accordingly lor _f_F can be expressed as the sum of two parts. The first part is the product of the matrices (lor _f_) F, lor _f_ being regarded as a 1 × 4 series matrix. The second part is that part of lor _f_F, in which the diffentiations operate upon the components of F alone. From 78) we obtain

_f_F = -F*_f_* - 2L;

hence the second part of lor _f_F = -(lor F*)_f_* + the part of -2 lor L, in which the differentiations operate upon the components of F alone. We thus obtain

lor S = (lor _f_)F - (lor F*)_f_* + N,

where N is the vector with the components

N_{_h_} = ½(∂_f₂₃_/∂_x__{_h_} F₂₃ + ∂_f₃₁_/∂_x__{_h_} F₃₁ +
∂_f₁₂_/∂_x__{_h_} F₁₂ + ∂_f₁₄_/∂_x__{_h_} F₁₄
+ ∂_f₂₄_/∂_x__{_h_} F₂₄ + ∂_f₃₄_/∂_x__{_h_} F₃₄
- ∂F₂₃/∂_x__{_h_} _f₂₃_ - ∂F₃₁/∂_x__{_h_} _f_₃₁ - ∂F₁₂/∂_x__{_h_}
_f₁₂_ - ∂F₁₄/∂_x__{_h_} _f₁₄_
- ∂F₂₄/∂_x__{_h_} _f₂₄_ - ∂F₃₄/∂_x__{_h_} _f₃₄_),

(_h_ = 1, 2, 3, 4)

By using the fundamental relations A) and B), 90) is transformed into the fundamental relation

(91) lor S = -_s_F + N.

In the limitting case ε = 1, μ = 1, _f_ = F, N vanishes identically.

Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of N,—

(92) N_{_h_} = - ½ Φ[=Φ]∂ε/∂_x__{_h_} - ½ ψ[=ψ]∂μ/∂_x__{_h_}
+ (εμ - 1)(Ω₁ ∂ω₁/∂_x__{_h_} + Ω₂ ∂ω₂/∂_x__{_h_} + Ω₃ ∂ω₃/∂_x__{_h_}
+ Ω₄ ∂ω₄/∂_x__{_h_})

for _h_ = 1, 2, 3, 4.

Now if we make use of (59), and denote the space-vector which has Ω₁, Ω₂, Ω₃ as the _x_, _y_, _z_ components by the symbol W, then the third component of 92) can be expressed in the form

(93) (εμ - 1)/√(1 - _u²_) (W ∂_u_/∂_x__{_h_}),

The round bracket denoting the scalar product of the vectors within it.

§ 14. The Ponderomotive Force.[28]

Let us now write out the relation K = lor S = -_s_F + N in a more practical form; we have the four equations

(94) K₁ = ∂X_{_x_}/∂_x_ + ∂X_{_y_}/∂_y_ + ∂X_{_y_}/∂_z_ -
∂X_{_t_}/∂_t_ = ρE_{_x_} + _s__{_y_}M_{_z_} - _s__{_z_}M_{_x_}

- ½ Φ[=Φ] ∂ε/∂_x_ - ½ ψ[=ψ]∂μ/∂_x_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_x_),

(95) K₂ = ∂Y_{_x_}/∂_x_ + ∂Y_{_y_}/∂_y_ + ∂Y_{_z_}/∂_z_ -
∂Y_{_t_}/∂_t_ = ρE_{_y_} + _s__{_z_}M_{_x_} - _s__{_x_}M_{_y_}

- ½ Φ[=Φ]∂ε/∂_y_ - ½ ψ[=ψ]∂μ/∂_y_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_y_),

(96) K₃ = ∂Z_{_x_}/∂_x_ + ∂Z_{_y_}/∂_y_ + ∂Z_{_z_}/∂_z_ -
∂Z_{_t_}/∂_t_ = ρE₂ + _s__{_x_}M_{_y_} - _s__{_y_}M₄

- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂_z_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_z_),

(97) (1/_i_)K₄ = ∂T_{_y_}/∂_x_ - ∂T_{_y_}/∂_y_ - ∂T_{_z_}/∂_z_ -
∂T_{_t_}/∂_t_ = _s__{_x_}E_{_x_} + _s__{_y_}E_{_y_} +
_s__{_z_}E_{_z_}

- ½ Φ[=Φ]∂ε/∂_t_ - ½ ψ[=ψ]∂μ/∂_t_ + (εμ - 1)/√(1 - _u²_)
(W∂_u_/∂_t_).

It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point _x_, _y_, _z_, _t_, it has got, _x_, _y_, _z_ components as the first three components of the space-time vector

K + (ωK)ω,

This vector is perpendicular to ω; the law of Energy finds its expression in the fourth relation.

The establishment of this opinion is reserved for a separate tract.

In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK = 0, and we obtain the ordinary equations in the theory of electrons.

Footnote 9:

_Vide_ Note 1.

Footnote 10:

Note 2.

Footnote 11:

_Vide_ Note 3.

Footnote 12:

_Vide_ Note 4.

Footnote 13:

Note 5.

Footnote 14:

See notes on § 8 and 10.

Footnote 15:

See note 9.

Footnote 16:

See Note.

Footnote 17:

Vide Note.

Footnote 18:

Just as beings which are confined within a narrow region surrounding a
point on a spherical surface, may fall into the error that a sphere is
a geometric figure in which one diameter is particularly distinguished
from the rest.

Footnote 19:

Einzelne stelle der Materie.

Footnote 20:

Vide Note.

Footnote 21:

_Vide_ note 13.

Footnote 22:

_Vide_ note 14.

Footnote 23:

_Vide_ note 15.

Footnote 24:

_Vide_ note 16.

Footnote 25:

_Vide_ note 17.

Footnote 26:

_Vide_ note 19.

Footnote 27:

_Vide_ note 18.

Footnote 28:

Vide note 40.