Page:Electromagnetic phenomena.djvu/14

 Aundefined, A2, A3, etc., formed by the centres of the particles in the moving system Σ, is obtained from Aundefined', A2', A3' , etc. by means of a deformation $$\left(\frac{1}{kl},\ \frac{1}{l},\ \frac{1}{l}\right)$$. According to what has been said in § 8, the centres will of themselves take these positions Aundefined, A2, A3, etc. if originally, before there was a translation, they occupied the positions Aundefined', A2', A3' , etc.

We may conceive any point P' in the space of the system Σ'  to be replaced by the above deformation, so that a definite point P of Σ corresponds to it. For two corresponding points P'  and P we shall define corresponding instants, the one belonging to P' , the other to P, by stating that the true time at the first instant is equal to the local time, as determined by (5) for the point P, at the second instant. By corresponding times for two corresponding particles we shall understand times that may be said to correspond, if we fix our attention on the centres A'  and A of these particles.

b. As regards the interior state of the atoms, we shall assume that the configuration of a particle A in Σ at a certain time may be derived by means of the deformation $$\left(\frac{1}{kl},\ \frac{1}{l},\ \frac{1}{l}\right)$$ from the configuration of the corresponding particle in Σ' , such as it is at the corresponding instant. In so far as this assumption relates to the form of the electrons themselves, it is implied in the first hypothesis of § 8.

Obviously, if we start from a state really existing in the system Σ' , we have now completely defined a state of the moving system Σ. The question remains however, whether this state will likewise be a possible one.

In order to judge this, we may remark in the first place that the electric moments which we have supposed to exist in the moving system and which we shall denote by $$\mathfrak{p}$$ will be certain definite functions of the coordinates x, y, z of the centres A of the particles, or, as we shall say, of the coordinates of the particles themselves, and of the time t. The equations which express the relations between $$\mathfrak{p}$$ on one hand and x, y, z, t on the other, may be replaced by other equations, containing the vectors $$\mathfrak{p}'$$ defined by (25) and the quantities x',y',z',t'  defined by (4) and (5). Now, by the above assumptions a and b, if in a particle A of the moving system, whose coordinates are x, y, z, we find an electric moment $$\mathfrak{p}$$ at the time t, or at the local time t', the vector $$\mathfrak{p}'$$ given by (26) will be the moment which exists in the other system at the true time t'  in a particle whose coordinates are x', y', z' . It appears in this way that the equations between $$\mathfrak{p}'$$, x', y', z', t'  are the same for both systems, the difference being only this, that for the system Σ'  without translation