Page:Lorentz Simplified1899.djvu/15

 As to the corresponding vibratory motions, we shall require that at corresponding times, i.e. for equal values of t", the configuration of S may always be got from that of S0 by the above mentioned dilatations. Then, it appears from (7) that $$\mathfrak{a}_{x},\mathfrak{a}_{y},\mathfrak{a}_{z}$$ will be, in both systems, the same functions of x", y", z", t", whence we conclude that the equations (Ie)—(IVe) can be satisfied by values of $$\mathfrak{F}_{x},\mathfrak{H}_{x}$$, etc., which are likewise, in S0 and in S, the same functions of x", y", z", t"''.

Always provided that we start from a vibratory motion in S0 that can really exist, we have now arrived at a motion in S, that is possible in so far as it satisfies the electromagnetic equations. The last stage of our reasoning will be to attend to the molecular forces. In S0 we imagine again, around one of the ions, the same small space, we have considered in § 7 and to which the molecular forces acting on the ion are confined; in the other system we shall now conceive the corresponding small space, i.e. the space that may be derived from the first one by applying to it the dilatations (6). As before, we shall suppose these spaces to be so small that in the second of them there is no necessity to distinguish the local times in its different parts; then we may say that in the two spaces there will be, at corresponding times, corresponding distributions of matter.

We have already seen that, in the states of equilibrium, the electric forces parallel to OX, OY, OZ, existing in S differ from the corresponding forces in S0 by the factors

$$\frac{1}{\varepsilon^{2}},\ \frac{1}{k\varepsilon^{2}}$$ and $$\frac{1}{k\varepsilon^{2}}$$.

From (Ve) it appears that the same factors come into play when we consider the part of the electric forces that is due to the vibrations. If, now, we suppose that the molecular forces are modified in quite the same way in consequence of the translation, we may apply the just mentioned factors to the components of the total force acting on an ion. Then, the imagined motion in S will be a possible one, provided that these same factors to which we have been led in examining the forces present themselves again, when we treat of the product of the masses and the accelerations.

According to our suppositions, the accelerations in the directions of OX, OY, OZ in S are resp. $$\tfrac{1}{k^{3}\varepsilon}, \tfrac{1}{k^{2}\varepsilon}$$ and $$\frac{1}{k^{2}\varepsilon}$$ times what they Proceedings Royal Acad. Amsterdam. Vol. I.