# Translation:Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies/Section I

Translation:Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies by Hendrik Lorentz, translated from German by Wikisource
Section I. The fundamental equations for a system of ions located in the aether.

## The fundamental equations for a system of ions located in the aether.

### The equations for the aether.

§ 5. When forming the equations of motion we will express all magnitudes in electromagnetic measure, and preliminarily use a coordinate system that is at rest in the aether. Now according to Maxwell, two kinds of deviations from the equilibrium state can exist in this medium. The deviation of first kind, which (among others) can be found in the vicinity of any charged body, we call the dielectric displacement; it is a vector quantity and may get the designation ${\displaystyle {\mathfrak {d}}}$[1]. It is solenoidally distributed in "pure" aether, i.e. in the spaces between the ions, and we have

 ${\displaystyle Div\ {\mathfrak {d}}=0.}$ (3)

We now want to assume, that aether exists in the space where an ion is located, and that a dielectric displacement can happen at this place, i.e. that the dielectric displacement caused by a single ion is extended over the interior of the other ions.

The charge of an ion we see as distributed over a certain space; the spatial density may be called ${\displaystyle \rho }$, and we want to assume, that this function steadily goes over to 0 when passing from the interior of the particle into the pure aether. In this assumption, that gives us the advantage that no discontinuities must be considered, there is no essential restriction. Because the charge distribution over a surface and a discontinuity of ${\displaystyle \rho }$ can be treated as limiting cases of states to which that assumption are applicable.

In the cases to be considered, ${\displaystyle \rho }$ is different from zero only in the interior of a very great number of small spaces which are completely mutually separated. Yet we can start with the general case, that an electric density exists in arbitrary great spaces. Since we think of the electric charges as always connected to ponderable matter, then this would correspond to a continuous distribution of matter.

Ponderable matter, which is not charged, has only to be considered by us, when it exerts molecular forces on the ions. Concerning the electric phenomena, it has no influence at all and everything happens, as if the space where it is located would only contain the aether.

Where ${\displaystyle \rho }$ is different form zero, equation (3) is not applicable anymore. According to a known theorem from Maxwell's theory, we have for any closed surface ${\displaystyle \sigma }$, when E represents the entire charge in the interior,

${\displaystyle \int {\mathfrak {d}}_{n}\ d\ \sigma =E=\int \rho \ d\ \tau {,}}$

or

${\displaystyle \int Div\ {\mathfrak {d}}\ d\ \tau =\int \rho \ d\ \tau {,}}$

so everywhere it must be

 ${\displaystyle Div\ {\mathfrak {d}}=\rho }$ (I)

If the ponderable matter is moving, then — since it carries the charge along with it — at a certain point of space there always exists another ${\displaystyle \rho }$, and soon it is (when we are dealing with mutually separated ions) different from zero here and there. Yet the condition of the aether has constantly to obey equation (I).

§ 6. The change of ${\displaystyle {\mathfrak {d}}}$, that happens with time at a certain point of space, constitutes an electric current (Maxwell's displacement current) that can be represented by ${\displaystyle {\dot {\mathfrak {d}}}}$. We assume that it exists also in the interior of charged matter. Yet additionally we find a convection current ${\displaystyle {\mathfrak {C}}}$ there. This is considered by me, when ${\displaystyle {\mathfrak {v}}}$ is the velocity of ponderable matter, as given in magnitude and direction by

${\displaystyle {\mathfrak {C}}=\rho {\mathfrak {v}}}$

and I put for the total current

 ${\displaystyle {\mathfrak {S}}={\mathfrak {C}}+{\mathfrak {\dot {d}}}=\rho {\mathfrak {v}}+{\dot {\mathfrak {d}}}.}$ (4)

In charged matter, ${\displaystyle {\mathfrak {v}}}$ shall continuously vary from point to point[2]. Additionally the charge of every mass element shall stay unchanged during the motion. Thus ${\displaystyle \rho \omega }$ must be constant, when ${\displaystyle \omega }$ is the — maybe variable — volume of the element.

From this assumption we derive the property of solenoid distribution for the total current, which will be expressed by

 ${\displaystyle Div\ {\mathfrak {S}}=0.}$ (5)

§ 7. The second deviation of the equilibrium state of the aether will be determined by the magnetic force ${\displaystyle {\mathfrak {H}}}$. It depends on the instantaneous current distribution, and satisfies the requirements

 ${\displaystyle Div\ {\mathfrak {H}}=0{,}}$ (II)
 ${\displaystyle Rot\ {\mathfrak {H}}=4\pi {\mathfrak {S}},}$ (III)

whose applicability we also presuppose for the interior of ponderable matter[3].

Eventually we also assume the relation, for the interior of the ions[4] as well as for the interspaces, by which in Maxwell's theory the dielectric displacement is connected with the temporal variation of the magnetic force. The relation reads

 ${\displaystyle -4\pi V^{2}Rot\ {\mathfrak {d}}={\dot {\mathfrak {H}}}{,}}$ (IV)

if we denote by V the ratio of the electromagnetic and electrostatic units of electricity, or the velocity of light in the aether.

Now we have written down all equations for the aether. If ${\displaystyle {\mathfrak {d}}}$ and ${\displaystyle {\mathfrak {H}}}$ for ${\displaystyle t=0}$ are given everywhere, we know for all subsequent instants the motion of charged bodies, and if we also add the requirement, that ${\displaystyle {\mathfrak {d}}}$ and ${\displaystyle {\mathfrak {H}}}$ vanish in infinite distance, then these vectors are definitely specified.

Where ${\displaystyle \rho =0}$, the equations go over into the formulas for pure aether, from which it is knowingly given, that the variations represented by ${\displaystyle {\mathfrak {d}}}$ and ${\displaystyle {\mathfrak {H}}}$ propagate with the velocity of light.

Since the equations are linear, various solutions can be composed to a more general one by addition. For example, the motion of n ions shall be given, and n value systems of ${\displaystyle {\mathfrak {d}}}$ and ${\displaystyle {\mathfrak {H}}}$ shall be found that determine the state of the aether for the case in which only one ion exists, and the others were neglected. Then we obtain by superposition the state of the aether, being in agreement with the motions of all n ions. In this sense we may say, that any ion influences the state of aether in exactly such way, as if the others wouldn't exist.

§ 8. If the ponderable matter is at rest and ${\displaystyle {\mathfrak {d}}}$ is independent of time, then ${\displaystyle {\mathfrak {S}}}$ and ${\displaystyle {\mathfrak {H}}}$ vanish, while ${\displaystyle {\mathfrak {d}}}$ will be determined by

 ${\displaystyle Div\ {\mathfrak {d}}=\rho }$ (I)

and

${\displaystyle Rot\ {\mathfrak {d}}=0.}$

This last equation says, that ${\displaystyle {\mathfrak {d}}_{x},\ {\mathfrak {d}}_{y},\ {\mathfrak {d}}_{z}}$ can be considered as partial derivatives of a single function, which we want to call ${\displaystyle -{\tfrac {\omega }{4\pi }}}$. We thus put

 ${\displaystyle {\mathfrak {d}}_{x}=-{\frac {1}{4\pi }}{\frac {\partial \omega }{\partial x}}{\text{, etc.}}}$ (6)

and derive from (I)

 ${\displaystyle \triangle \omega =-4\pi \rho .}$ (7)

After we determined ${\displaystyle \omega }$ from that, ${\displaystyle {\mathfrak {d}}_{x},\ {\mathfrak {d}}_{y},\ {\mathfrak {d}}_{z}}$ can be calculated from (6).

### The first part of the force acting on ponderable matter.

§ 9. According to the older electrostatics, whose conclusions are in agreement with experience, we obtain the force components that act on the volume element in the case previously considered, when we at first determine the "potential function" by means of Poisson's equation, and then multiply its derivatives by ${\displaystyle -V^{2}\rho \ d\ \tau }$[5]. Since our formula (7) is in agreement with Poisson's equations, the potential function must coincide with ${\displaystyle \omega }$; therefore we have to assume as values of the force components

 ${\displaystyle -V^{2}\rho {\frac {\partial \omega }{\partial x}}\ d\ \tau {\text{, etc.}}}$ (8)

If the forces, as it is claimed by Maxwell's theory, shall be caused by the state of the aether, then it is probable that it depends on the dielectric displacement in the considered volume element. Indeed, when we consider (6), we can write for (8)

${\displaystyle 4\pi V^{2}{\mathfrak {d}}_{x}\rho \ d\ \tau {\text{, etc.}}}$

Therefore I will assume, that in all cases in which a dielectric displacement exists in element ${\displaystyle d\tau }$, the aether exerts a force with the mentioned components on ponderable matter located at this place, i.e. a force[6] which can be represented for the unit of charge by

${\displaystyle {\mathfrak {E}}_{1}=4\pi V^{2}{\mathfrak {d}}.}$

§ 10. Let two stationary ions with charges e and ${\displaystyle e'}$ be given, whose dimensions are small in relation to distance r. To find the force that acts on the first one, we have to decompose it into space elements, to apply on any of them the previous theorem, and then to integrate. Thereby ${\displaystyle {\mathfrak {d}}}$ may be considered as composed of the dielectric displacements, that stem from the first and the second particle. We easily find that the first part of ${\displaystyle {\mathfrak {d}}}$ doesn't contribute anything to the total force. The second part has (within the first ion) everywhere the direction of r and the magnitude ${\displaystyle e'/4\pi r^{2}}$; so e will be repulsed by ${\displaystyle e'}$ by the force

${\displaystyle V^{2}{\frac {ee'}{r^{2}}}.}$

As this is in agreement with Coulomb's laws, it is clear that the theory of ions, as regards the ordinary problems, leads back to the older way of treatment.

### Electric currents in ponderable conductors.

§ 11. In a ponderable conductor, in which a current flows through, and in which innumerable ions are in motion according to our view, ${\displaystyle {\mathfrak {d}}}$, ${\displaystyle {\mathfrak {S}}}$ and ${\displaystyle {\mathfrak {H}}}$ are changing in an irregular way from point to point. Yet from equations (II) and (III) it follows

${\displaystyle {\begin{array}{l}Div\ {\bar {\mathfrak {H}}}=0{,}\\Rot\ {\bar {\mathfrak {H}}}=4\pi {\bar {\mathfrak {S}}}{;}\\\end{array}}}$

since ${\displaystyle {\bar {\mathfrak {H}}}}$ coincides with ${\displaystyle {\mathfrak {H}}}$ in measurable distance from the conductor, and the action into the outside will only be determined by the average current ${\displaystyle {\bar {\mathfrak {S}}}}$. It is this current, with which the ordinary theory (which neglects molecular processes) is dealing.

By equation (4) we have

${\displaystyle {\bar {\mathfrak {S}}}={\overline {\rho {\mathfrak {v}}}}+{\dot {\bar {\mathfrak {d}}}}.}$

If the state of flow is stationary, then the observable magnitudes and also the averages are independent of time. Thus it will be

${\displaystyle {\bar {\mathfrak {S}}}={\overline {\rho {\mathfrak {v}}}}{,}}$

i.e. only the convection currents cause the action into the outside.

By the definition given in § 4, the components of ${\displaystyle {\overline {\rho {\mathfrak {v}}}}}$ are

${\displaystyle {\frac {1}{I}}\int \rho {\mathfrak {v}}_{x}\ d\ \rho {\text{, u. s. w.,}}}$

or, when ${\displaystyle \rho }$ is different from zero only within the ions, and any ion is displaced without rotation

${\displaystyle {\frac {1}{I}}\sum e{\mathfrak {v}}{\text{, etc.,}}}$

where e is the charge of an ion, and the sum is related to all charged particles contained in sphere I. We can easily see, that the result can be summarized in the formula

${\displaystyle {\bar {\mathfrak {S}}}={\frac {1}{I}}\sum e{\mathfrak {v}}}$

and this remains valid, when we don't interpret I just as a sphere, but as an arbitrary space, whose dimensions (albeit very small) are nevertheless much greater than the average distance of the ions. Of course, then the sum must be extended over the chosen space as well.

If there is a current within a lead wire with cross-section ${\displaystyle \omega }$, then we can take for I the part, that lies between two cross-sections which are mutually distant by ds[7]. Since the magnitude of current will be determined by:

${\displaystyle i=\omega {\bar {\mathfrak {S}}}{,}}$

and ${\displaystyle I=\omega \ d\ s}$, thus we obtain

${\displaystyle \sum e{\mathfrak {v}}=i\ d\ s{,}}$

where ${\displaystyle i\ d\ s}$ is to be considered as a vector in the direction of the current.

### The second part of the force acting on ponderable matter

§ 12. A current element as the one previously considered, may be located in a magnetic field generated by external causes. According to a known law it suffers an electrodynamic force

${\displaystyle [i\ d\ s.{\mathfrak {H}}]{,}}$

for which we also can write now

${\displaystyle \left[\sum e{\mathfrak {v}}.{\mathfrak {H}}\right]{,}}$

or

${\displaystyle \sum \{e[{\mathfrak {v}}.{\mathfrak {H}}]\}.}$

This action results (according to our view) from the forces, which will be exerted by the aether upon the ions of the current element. It is thus near at hand, to assume for the force acting on a single ion

${\displaystyle e[{\mathfrak {v.H}}]{,}}$

a hypothesis, which we still want to extend in a way, so that we generally assume a force acting on ponderable matter of the volume element ${\displaystyle d\tau }$

${\displaystyle \rho d\ \tau [{\mathfrak {v.H}}]}$

In unit charge this would be

${\displaystyle {\mathfrak {E}}_{2}=[{\mathfrak {v.H}}]}$[8].

By putting this vector together with ${\displaystyle {\mathfrak {E}}_{1}}$ that was considered earlier (§ 9), we obtain for the total force exerted on the unit of charge, i.e. for the electric force,

 ${\displaystyle {\mathfrak {E}}=4\pi V^{2}{\mathfrak {d}}+[{\mathfrak {v.H}}].}$ (V)

We refuse to express the thus stated law by words. By elevating it to a general fundamental-law, we have completed the system of equations of motion (I)—(V), since the electric force, in connection with possible other forces, determines the motion of ions.

Concerning the latter, we still want to introduce the assumption, that the ions never rotate.[9]

### The conservation of energy.

§ 13. To justify our hypotheses, it is necessary to show its agreement with the energy law. We consider an arbitrary system of ponderable bodies that contain ions, around which only the aether exists up to an infinite distance, and around it we put an arbitrarily closed surface ${\displaystyle \sigma }$. During an element of time ${\displaystyle dt}$, the work that affects ponderable matter and which stems from ${\displaystyle {\mathfrak {E}}}$, is

${\displaystyle 4\pi V^{2}d\ t\int \rho \left({\mathfrak {d}}_{x}{\mathfrak {v}}_{x}+{\mathfrak {d}}_{y}{\mathfrak {v}}_{y}+{\mathfrak {d}}_{z}{\mathfrak {v}}_{z}\right)d\ \tau {,}}$

where it is to be noted, that no work is done by the forces (which are derived from ${\displaystyle {\mathfrak {E}}_{2}}$), because they are always perpendicular to the direction of motion. Furthermore, if dA is the work of all other forces acting on matter, and L is the ordinary mechanical energy of that matter, then

 ${\displaystyle d\ A=d\ L-4\pi V^{2}d\ t\int \rho \left({\mathfrak {d}}_{x}{\mathfrak {v}}_{x}+{\mathfrak {d}}_{y}{\mathfrak {v}}_{y}+{\mathfrak {d}}_{z}{\mathfrak {v}}_{z}\right)d\ \tau .}$ (9)

The integral is related to the space filled with ponderable matter; but we can also extend it over the entire space enclosed by ${\displaystyle \sigma }$. All other space integrals in this § are to be understood in the latter sense.

We replace in (9), by (4) and (III)

${\displaystyle 4\pi \rho {\mathfrak {v}}_{x}{\text{. u. s. w.}}}$

by

 ${\displaystyle {\frac {\partial {\mathfrak {H}}_{z}}{\partial y}}-{\frac {\partial {\mathfrak {H}}_{y}}{\partial z}}-4\pi {\frac {\partial {\mathfrak {d}}_{x}}{\partial t}}{\text{, u. s. w.,}}}$ (10)

and transform the parts of the integral, that contain derivatives of ${\displaystyle {\mathfrak {H}}_{x},{\mathfrak {H}}_{y},{\mathfrak {H}}_{z}}$, by partial integration.

By consideration of equation (IV) we will find

 ${\displaystyle d\ A=d(L+U)+V^{2}d\ t\int \ [{\mathfrak {d.H}}]_{n}d\ \sigma {,}}$ (11)

where

 ${\displaystyle U=2\pi V^{2}\int \ {\mathfrak {d}}^{2}d\ \tau +{\frac {1}{8\pi }}\int {\mathfrak {H}}^{2}d\ \tau }$ (12)

At first is should be assumed, that the electric motions are restricted to a certain finite space, and that surface ${\displaystyle \sigma }$ is entirely outside of that space. Then at the surfaces it will be ${\displaystyle {\mathfrak {d}}=0,{\mathfrak {H}}=0}$, and

${\displaystyle d\ A=d(L+U).}$

Therefore the magnitude L + U really applies, whose increase is equal to the work of the external forces, and which therefore is denoted by the expression "energy". It is composed of the ordinary mechanical energy L and the "electrical" energy U, and as regards the latter we find again the value given by Maxwell.

### The theorem of Pointing.

§ 14. Even if we abandon the previously made assumption about ${\displaystyle \sigma }$, formula (11) allows of a simple interpretation. With Maxwell we not only assume that the electric force would have the value (12), but also, that it is really distributed over the space as it is expressed by the formula, i.e. that it amounts for unit volume

${\displaystyle 2\pi V^{2}{\mathfrak {d}}^{2}+{\frac {1}{8\pi }}{\mathfrak {H}}^{2}}$

In equations (11), L + U thus means the whole energy within surface ${\displaystyle \sigma }$, and therefore the view is near at hand, that a quantity of energy

${\displaystyle V^{2}d\ t\int \ [{\mathfrak {d.H}}]_{n}d\ \sigma }$

has traveled through the surface into the outside. It is most simple, if we put for the "energy flow" related to unit time and area

 ${\displaystyle V^{2}[{\mathfrak {d.H}}]_{n}.}$ (13)

By that we come to the known theorem formulated by Poynting. Here, we don't discuss the subtle, related question concerning the localization of the energy. We can restrict ourselves with the fact, that the entire energy located in an arbitrary space — the "electric" portion calculated by formula (12) — always varies, as if the energy would travel according to the way determined by (13).

### Tensions in the aether.

§ 15. The forces determined by our formula (V), not only require the motion of ions in ponderable bodies, but also in some circumstances can unify themselves to an action, that tends to set the body into motion. In this way all "ponderomotive" forces emerge, as for example the ordinary electrostatic and electrodynamic ones, as well as the pressure that is exerted by light rays on a body.

We want to consider the body as rigid, and calculate (by simple addition) all the forces that were exerted by the aether in the direction of the x-axis, i.e. the total force ${\displaystyle \Xi }$ in this direction. The investigation should be based on the things said at the beginning of § 13.

We immediately obtain

${\displaystyle {\begin{array}{cl}\Xi &=4\pi V^{2}\int {\mathfrak {d}}_{x}\rho \ d\ \tau +\int \rho [{\mathfrak {v.H}}]_{x}\ d\ \tau =\\&=4\pi V^{2}\int {\mathfrak {d}}_{x}\rho \ d\ \tau +\int \rho \left({\mathfrak {v}}_{y}{\mathfrak {H}}_{z}-{\mathfrak {v}}_{z}{\mathfrak {H}}_{y}\right)\ d\ \tau {,}\end{array}}}$

where the integrals only have to be extended over the ponderable body, but like in § 13, it should taken for the entire space enclosed by ${\displaystyle \sigma }$.

At first, we replace ${\displaystyle 4\pi \rho {\mathfrak {v}}_{x}}$, etc. by the expressions (10), and, because of (I), ${\displaystyle \rho }$ by

${\displaystyle {\frac {\partial {\mathfrak {d}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {d}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {d}}_{z}}{\partial z}}}$

thus

 ${\displaystyle {\begin{array}{c}\Xi =4\pi V^{2}\int {\mathfrak {d}}_{x}\left({\frac {\partial {\mathfrak {d}}_{x}}{\partial x}}+{\frac {\partial {\mathfrak {d}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {d}}_{z}}{\partial z}}\right)\ d\ \tau +\\+{\frac {1}{4\pi }}\int \left\{{\mathfrak {H}}_{z}\left({\frac {\partial {\mathfrak {H}}_{x}}{\partial z}}-{\frac {\partial {\mathfrak {H}}_{z}}{\partial x}}\right)-{\mathfrak {H}}_{y}\left({\frac {\partial {\mathfrak {H}}_{y}}{\partial x}}-{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}\right)\right\}\ d\ \tau +\\+\int \left({\mathfrak {H}}_{y}{\frac {\partial {\mathfrak {d}}_{z}}{\partial t}}-{\mathfrak {H}}_{z}{\frac {\partial {\mathfrak {d}}_{y}}{\partial t}}\right)\ d\ \tau .\end{array}}}$ (14)

Furthermore, a partial integration and application of (IV) and (II) gives (when we denote the direction constants of the perpendicular to ${\displaystyle \sigma }$ by ${\displaystyle \chi ,\beta ,\gamma }$)

${\displaystyle {\begin{array}{cl}\int {\mathfrak {d}}_{x}{\frac {\partial {\mathfrak {d}}_{y}}{\partial y}}\ d\ \tau &=\int \beta {\mathfrak {d}}_{x}{\mathfrak {d}}_{y}\ d\ \sigma -\int {\mathfrak {d}}_{y}{\frac {\partial {\mathfrak {d}}_{x}}{\partial y}}\ d\ \tau =\\&=\int \beta {\mathfrak {d}}_{x}{\mathfrak {d}}_{y}\ d\ \sigma -\int {\mathfrak {d}}_{y}{\frac {\partial {\mathfrak {d}}_{y}}{\partial x}}\ d\ \tau -{\frac {1}{4\pi V^{2}}}\int {\mathfrak {d}}_{y}{\frac {\partial {\mathfrak {H}}_{z}}{\partial t}}\ d\ \tau {,}\\\int {\mathfrak {d}}_{x}{\frac {\partial {\mathfrak {d}}_{z}}{\partial z}}\ d\ \tau &=\int \gamma {\mathfrak {d}}_{x}{\mathfrak {d}}_{z}\ d\ \sigma -\int {\mathfrak {d}}_{z}{\frac {\partial {\mathfrak {d}}_{x}}{\partial z}}\ d\ \tau =\\&=\int \gamma {\mathfrak {d}}_{x}{\mathfrak {d}}_{z}\ d\ \sigma -\int {\mathfrak {d}}_{z}{\frac {\partial {\mathfrak {d}}_{z}}{\partial x}}\ d\ \tau +{\frac {1}{4\pi V^{2}}}\int {\mathfrak {d}}_{z}{\frac {\partial {\mathfrak {H}}_{y}}{\partial t}}\ d\ \tau {,}\end{array}}}$

${\displaystyle \int \left({\mathfrak {H}}_{y}{\frac {\partial {\mathfrak {H}}_{x}}{\partial y}}+{\mathfrak {H}}_{z}{\frac {\partial {\mathfrak {H}}_{x}}{\partial z}}\right)\ d\ \tau =\int \left(\beta {\mathfrak {H}}_{x}{\mathfrak {H}}_{y}+\gamma {\mathfrak {H}}_{x}{\mathfrak {H}}_{z}\right)\ d\ \sigma -}$

${\displaystyle -\int {\mathfrak {H}}_{x}\left({\frac {\partial {\mathfrak {H}}_{y}}{\partial y}}+{\frac {\partial {\mathfrak {H}}_{z}}{\partial z}}\right)\ d\ \tau =\int \left(\beta {\mathfrak {H}}_{x}{\mathfrak {H}}_{y}+\gamma {\mathfrak {H}}_{x}{\mathfrak {H}}_{z}\right)\ d\ \sigma +}$

${\displaystyle +\int {\mathfrak {H}}_{x}{\frac {\partial {\mathfrak {H}}_{x}}{\partial x}}\ d\ \tau .}$

If we substitute this value into (14), then several terms occur, that can be completely integrated, and eventually by a simple transformation we have

 ${\displaystyle \Xi =2\pi V^{2}\int \left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)\ d\ \sigma +{\frac {1}{8\pi }}\int \left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right)\ d\ \sigma +}$ ${\displaystyle +{\frac {d}{d\ t}}\int \left({\mathfrak {H}}_{y}{\mathfrak {d}}_{z}-{\mathfrak {H}}_{z}{\mathfrak {d}}_{y}\right)\ d\ \tau {,}}$ (15)

Two similar equations serve for the determination of the other components ${\displaystyle \mathrm {H} }$ and ${\displaystyle \mathrm {Z} }$ of the ponderomotive action.

Besides it is to be noticed, that ${\displaystyle \Xi }$, ${\displaystyle \mathrm {H} }$ and ${\displaystyle \mathrm {Z} }$ must vanish, as soon space ${\displaystyle \tau }$ doesn't contain ponderable matter. Then it would be

 ${\displaystyle 2\pi V^{2}\int \left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)\ d\ \sigma +{\frac {1}{8\pi }}\int \left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right)\ d\ \sigma =}$ ${\displaystyle =-{\frac {d}{d\ t}}\int \left({\mathfrak {H}}_{y}{\mathfrak {d}}_{z}-{\mathfrak {H}}_{z}{\mathfrak {d}}_{y}\right)\ d\ \tau {\text{, u. s. w.}}}$ (16)

§ 16. In some cases the space integral the remained in (15), will become independent of t, and if the last member vanishes, namely as soon as we have to deal with a stationary state, may it be with an electric charge, or may it be with a system of constant currents. Then, at least concerning the resultant force, the ponderomotive action can be calculated by integration over an arbitrary surface that encloses the body, and it is near at hand, to view them in a way, so that we (like Maxwell did) attribute to the aether a certain state of tension, and consider the tensions as the cause of the ponderomotive actions.[10] If we as usual understand by ${\displaystyle \left(X_{n},Y_{n},Z_{n}\right)}$ the force related to unit area, that the aether exerts at the side (given by n) of an element ${\displaystyle d\sigma }$ upon the opposite aether, then by (15) we would have to put

 ${\displaystyle X_{n}=2\pi V^{2}\left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)+{\frac {1}{8\pi }}\left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right){\text{, etc.}}}$ (17)

From that, it is easy to derive the values of ${\displaystyle X_{x}}$, ${\displaystyle X_{y}}$, ${\displaystyle X_{z}}$, ${\displaystyle Y_{x}}$; then we exactly obtain the system of tensions that was given by Maxwell.

§ 17. Since in (15) the space integral doesn't vanish, the assumption of tensions (17) doesn't generally lead to the action stated by us. If we would reject equation (V) as the basis of the calculation of the ponderomotive forces, and employ the tensions, then the case would in no way be finished by formulas (I)—(IV) and (17). One wouldn't even obtain the same values for ${\displaystyle \Xi }$, when we would apply the equation

${\displaystyle \Xi =2\pi V^{2}\int \left(2{\mathfrak {d}}_{x}{\mathfrak {d}}_{n}-\alpha {\mathfrak {d}}^{2}\right)\ d\ \sigma +{\frac {1}{8\pi }}\int \left(2{\mathfrak {H}}_{x}{\mathfrak {H}}_{n}-\alpha {\mathfrak {H}}^{2}\right)\ d\ \sigma }$

on one area and then on the other area, that encloses the considered body. It is connected with the fact, that the tensions (17) wouldn't let the aether to be at rest.

Above we have found formulas (16) for a space which is free of ponderable matter. That it's correct, as long as the aether is at rest, can hardly be doubted, since for the derivation only generally taken equations come into play. From the formulas

${\displaystyle Div\ {\mathfrak {d}}=0}$

and

${\displaystyle Rot\ {\mathfrak {H}}=4\pi {\dot {\mathfrak {d}}}}$

it is given, namely, that the right-hand side of equation (14) is zero for the free aether; the application of (IV) and (II) then leads to the first of formulas (16).

Now, in those formulas, the forces (that follow from the tensions at surface ${\displaystyle \sigma }$) are on the left side, and thus the formulas say that the considered part of the aether cannot remain at rest under the influence of these forces. All who consider equations (17) as generally valid, must conclude that in all cases, where Poynting's energy flow is variable with time[11], the aether as a whole will be set into motion. Thus it would be necessary to study the from of the emerging aether flows, and under consideration of them to again tackle the question after the ponderomotive actions.

The basics of a theory of the mentioned aether flows was already sketched by the masterhand of Hermann von Helmholtz in one[12] of his last papers, which he was able to complete.

We cannot discuss the considered questions, because the fundamental assumption by which we started, gives another view. Indeed, why should we, since we assumed that the aether is not in motion, ever speak about a force acting on that medium? The most simple would by, to assume that on a volume element of the aether, considered as a whole, never acts a force, or even refuse to apply the concept of force on such an element that of course never moves from its place. Of course this view violates the equality of action and reaction —, since we have reason to say that the aether exerts forces on ponderable matter —; but, as far I can see, nothing forces us to elevate this theorem to an unrestricted fundamental law.

Once we have decided ourselves in favor of the previously discusses view, then we must refuse from the outset, to reduce the ponderomotive forces (that follow from (V)) to tensions of the aether.

Nevertheless we may apply equation (15) to simplify the calculation, and it won't cause a misunderstanding, when we express ourselves for brevities sake, as if the elements of the two first integrals would mean real tensions in the aether.

From these merely fictitious "tensions" we can, as we saw, directly derive the interaction between charged bodies and electrodynamic actions. It is also to be recommended, to operate with them, when the phenomena are periodic and when we only wish to know the averages of the ponderomotive forces during a full period; the last member of (15) namely doesn't contribute anything to these values.

In this way we come to Maxwell's theorem on the pressure generated by motion of light.

### The reversibility of motions and the mirror image of motion.

§ 18. For subsequent applications we include the following considerations at this place.

Let a system of moving ions be given, and ${\displaystyle \rho _{1}}$, ${\displaystyle {\mathfrak {v}}_{1}}$, ${\displaystyle {\mathfrak {d}}_{1}}$ and ${\displaystyle {\mathfrak {H}}_{1}}$ are the various relevant magnitudes within. We may denote the corresponding magnitudes for a second system by ${\displaystyle \rho _{2}}$, ${\displaystyle {\mathfrak {v}}_{2}}$, ${\displaystyle {\mathfrak {d}}_{2}}$ and ${\displaystyle {\mathfrak {H}}_{2}}$, and we want to imagine that in an arbitrary point, these magnitudes are at time +t in agreement with the magnitudes ${\displaystyle \rho _{1}}$, ${\displaystyle -{\mathfrak {v}}_{1}}$, ${\displaystyle {\mathfrak {d}}_{1}}$ and ${\displaystyle -{\mathfrak {H}}_{1}}$ at time -t.

We can easily see that, as regards ${\displaystyle \rho _{2}}$ and ${\displaystyle {\mathfrak {v}}_{2}}$, those conditions can be satisfied by a real motion of the ions, and namely the system of these ions must completely be in agreement with the first system; the same configurations with the same interval must occur one after the other, as in that first system, but in opposite order; in other words, we obtain the motions of the ions in the second system, when we reverse the motions given at first.

Furthermore, since ${\displaystyle {\mathfrak {d}}_{2}}$ and ${\displaystyle {\mathfrak {H}}_{2}}$ satisfy the conditions (I), (II), (III) and (IV), thus the condition of the aether as determined by these vectors, is in agreement with the motion of the ions.

Eventually it follows from equation (V), that in the second system at time +t, the forces exerted upon the ions have the same direction and magnitude, as the corresponding forces in the first system at time -t. Now, if also the remaining forces that act on the ions in both cases — and in the same instances — are the same, then we can conclude, that the second state of motion is realizable in any way.

By means of similar considerations the possibility of motion can be demonstrated, which is the "mirror image" of a given motion with respect to a fixed plane.

We call ${\displaystyle P_{2}}$ the mirror image of a point ${\displaystyle P_{1}}$ and denote the magnitudes that are valid for two system — namely for the first in ${\displaystyle P_{1}}$ and for the second in ${\displaystyle P_{2}}$ — by ${\displaystyle \rho _{1}}$, ${\displaystyle {\mathfrak {v}}_{1}}$, ${\displaystyle {\mathfrak {d}}_{1}}$, ${\displaystyle {\mathfrak {H}}_{1}}$ and ${\displaystyle \rho _{2}}$, ${\displaystyle {\mathfrak {v}}_{2}}$, ${\displaystyle {\mathfrak {d}}_{2}}$, ${\displaystyle {\mathfrak {H}}_{2}}$. There it should constantly be ${\displaystyle \rho _{2}=\rho _{1}}$, and the vectors ${\displaystyle {\mathfrak {v}}_{2}}$, ${\displaystyle {\mathfrak {d}}_{2}}$, ${\displaystyle {\mathfrak {H}}_{2}}$ should be the mirror images of the vectors ${\displaystyle {\mathfrak {v}}_{1}}$, ${\displaystyle {\mathfrak {d}}_{1}}$ and ${\displaystyle -{\mathfrak {H}}_{1}}$.

That the second state of motion can now conveniently be called "mirror image", requires no explanation. If the forces of non-electric origin are of such a manner, so that the vectors by which they can be represented in both cases behave like objects and their mirror images, then the second motion will be possible as soon as the first one is possible.

1. A prove of the designations employed can be found at the end of the treatise.
2. By that it is of course not excluded, that mutually separated ions can often have very different velocities.
3. The justification of this lies in equation (5)
4. We neglect special magnetic properties of ponderable matter — which by the way would be explained by the motion of ions. Consequently we don't have to distinguish between the magnetic force and the magnetic induction.
5. The factor ${\displaystyle V^{2}}$ must be added, because we use the electromagnetic system of units.
6. Since this force is the only one, which exists in relation to electrostatic phenomena, it can well be called electrostatic force, although in general it also depends on the motion of ions.
7. Here, this letter doesn't mean something infinitely small in the strict sense of the word, but a distance that is of course very small compared to the dimensions of the conductor, but nevertheless much greater than the distance of the molecules.
8. If we don't want to consider an ordinary electric current as a convection current, then we must substantiate this formula by the assumption, that a body in which a convection takes place, experiences the same electrodynamic actions as a corresponding current conductor.
9. In an earlier published derivation of the equations of motion (La théorie électromagnétique de Maxwell et son application aux corps mouvants), I have discussed the necessary conditions.
10. Also with respect to the resultant force couple, the ponderomotive action on a rigid body is equivalent to the system of tensions (17) on an arbitrary surface ${\displaystyle \sigma }$ that encloses the body. If we also want to consider the ponderomotive actions on flexible or fluid bodies, then we would have to come back to volume elements. But this would lead too far.
11. Except the factor ${\displaystyle -V^{2}}$, the components of the energy flow are located on the right-hand side of equations (16) under the integral sign.
12. v. Helmholtz. Folgerungen aus Maxwell’s Theorie über die Bewegungen des reinen Aethers. Berl. Sitz.-Ber., 5. Juli 1893; Wied. Ann., Bd. 53, p. 135, 1894.