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Since these equations do no longer explicitly contain the velocity ${\mathfrak {p}}_{x}$, they will hold, without any change of form, for a system that has no translation, in which case, of course, t' would be the same thing as the universal time t.

Yet, strictly speaking, there would be a slight difference in the formulae, when applied to the two cases. In the system without a translation ${\mathfrak {a}}_{x},{\mathfrak {a}}_{y},{\mathfrak {a}}_{z}$ would be, in all points of an ion, the same functions of t', i. e. of the universal time, whereas, in the moving system, these components would not depend in the same way on t' in different parts of the ion, just because they must everywhere be the same functions of t.

However, we may ignore this difference, of the ions are so small, that we may assign to each of them a single local time, applicable to all its parts.

The equality of form of the electromagnetic equations for the two cases of which we have spoken will serve to simplify to a large extent our investigation. However, it should be kept in mind, that, to the equations (I_d)—(IV_d), we must add the equations of motion for the ions themselves. In establishing these, we have to take into account, not only the electric forces, but also all other forces acting on the ions. We shall call these latter the molecular forces and we shall begin by supposing them to be sensible only at such small distances, that two particles of matter, acting on each other, may be said to have the same local time.

§ 7. Let us now imagine two systems of ponderable bodies, the one S with a translation, and the other one S₀ without such a motion, but equal to each other in all other respects. Since we neglect quantities of the order ${\mathfrak {p}}_{x}^{2}/V^{2}$, the electric force will, by §5 be the same in both systems, as long as there are no vibrations.

After these have been excited, we shall have for both systems the equations (I_d)—(IV_d).

Further we shall imagine motions of such a kind, that, if in a point (x',y',z') of S₀ we find a certain quantity of matter or a certain electric charge at the universal time t, an equal quantity of matter or an equal charge will be found in the corresponding point of S at the local time t'. Of course, this involves that at these corresponding times we shall have, in the point (x', y', z') of both systems, the same electric density, the same displacement ${\mathfrak {a}}$, and equal velocities and accelerations.

Thus, some of the dependent variables in our equations (I_d)—(IV_d) will be represented in S₀ and S by the same functions of x', y', z', t',