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will lead us to the same result that Mossotti attained by means of the supposed inequality of the attractive and the repulsive forces.

§ 6. I shall suppose that each of the two disturbances of the aether is propagated with the velocity of light, and, taken by itself, obeys the ordinary laws of the electromagnetic field. These laws are expressed in the simplest form if, besides the dielectric displacement ${\mathfrak {d}}$, we consider the magnetic force ${\mathfrak {H}}$, both together determining, as we shall now say, one state of the aether or one field. In accordance with this, I shall introduce two pairs of vectors, the one ${\mathfrak {d}}$, ${\mathfrak {H}}$ belonging to the field that is produced by the positive ions, whereas the other pair ${\mathfrak {d}}'$, ${\mathfrak {H}}'$ serve to indicate the state of the aether which is called into existence by the negative ions. I shall write down two sets of equations, one for ${\mathfrak {d}}$, ${\mathfrak {H}}$, the other for ${\mathfrak {d}}',{\mathfrak {H}}'$, and having the form which I have used in former papers^[1] for the equations of the electromagnetic field, and which is founded on the assumption that the ions are perfectly permeable to the aether and that they can be displaced without dragging the aether along with them.

I shall immediately take this general case of moving particles.

Let us further suppose the charges to be distributed with finite volume-density, and let the units in which these are expressed be chosen in such a way that, in a body which exerts no electrical actions, the total amount of the positive charges has the same numerical value as that of the negative charges.

Let $\varrho$ be the density of the positive, and $\varrho '$ that of the negative charges, the first number being positive and the second negative.

Let ${\mathfrak {v}}$ (or ${\mathfrak {v}}'$) be the velocity of an ion.

Then the equations for the state $({\mathfrak {d}},{\mathfrak {H}})$ are^[2]

$\left.{\begin{array}{c}Div\ {\mathfrak {d}}=\varrho \\Div\ {\mathfrak {H}}=0\\Rot\ {\mathfrak {H}}=4\pi \varrho {\mathfrak {v}}+4\pi {\mathfrak {\dot {d}}}\\4\pi V^{2}Rot\ {\mathfrak {d}}=-{\mathfrak {\dot {H}}};\end{array}}\right\}$

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1. Lorentz. La théorie electromagnetique de Maxwell et son application aux corps mouvants, Arch. Néerl. XXV, p. 363; Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern.
2. $Div\ {\mathfrak {d}}={\frac {\partial {\mathfrak {d}}_{\mathsf {x}}}{\partial x}}+{\frac {\partial {\mathfrak {d}}_{\mathsf {y}}}{\partial y}}+{\frac {\partial {\mathfrak {d}}_{\mathsf {z}}}{\partial z}}$
   $Rot\ {\mathfrak {d}}$ is a vector, whose components are ${\frac {\partial {\mathfrak {d}}_{\mathsf {z}}}{\partial y}}-{\frac {\partial {\mathfrak {d}}_{\mathsf {y}}}{\partial z}}$ etc.