Page:Electromagnetic phenomena.djvu/13

direction of motion, the electron behaves as if it had a mass m₁, those in which the acceleration is normal to the path, as if the mass were m₂. These quantities m₁ and m₂ may therefore properly be called the "longitudinal" and "transverse" electromagnetic masses of the electron. I shall suppose that there is no other, no "true" or "material" mass.

Since k and l differ from unity by quantities of the order ${\frac {w^{2}}{c^{2}}}$, we find for very small velocities

$m_{1}=m_{2}={\frac {e^{2}}{6\pi c^{2}R}}$.

This is the mass with which we are concerned, if there are small vibratory motions of the electrons in a system without translation. If, on the contrary, motions of this kind are going on in a body moving with the velocity w in the direction of the axis of x, we shall have to reckon with the mass m₁, as given by (30), if we consider the vibrations parallel to that axis, and with the mass m₂, if we treat of those that are parallel to OY or OZ. Therefore, in short terms, referring by the index Σ to a moving system and by Σ' to one that remains at rest,

$m(\Sigma )=\left({\frac {d(klw)}{dw}},\ kl,\ kl\right)m(\Sigma ')$.

(31)

§ 10. We can now proceed to examine the influence of the Earth's motion on optical phenomena in a system of transparent bodies. In discussing this problem we shall fix our attention on the variable electric moments in the particles or "atoms" of the system. To these moments we may apply what has been said in § 7. For the sake of simplicity we shall suppose that, in each particle, the charge is concentrated in a certain number of separate electrons, and that the "elastic" forces that act on one of these and, conjointly with the electric forces, determine its motion, have their origin within the bounds of the same atom.

I shall show that, if we start from any given state of motion in a system without translation, we may deduce from it a corresponding state that can exist in the same system after a translation has been imparted to it, the kind of correspondence being as specified in what follows.

a. Let A', A₂', A₃' , etc. be the centres of the particles in the system without translation (Σ'); neglecting molecular motions we shall take these points to remain at rest. The system of points