Page:ComstockInertia.djvu/11

motion causes in the velocity of propagation through the system of electrical disturbances. This is seen in the simple case of a moving electron where the crowding of the lines towards the equator with increase of velocity is only partly due to the added energy. It is evident, therefore, that for velocities so great that the second order terms cannot be neglected, the mass depends on complicated terms which vary with the internal structure and motions of the system, and it does not appear as if a general expression for the mass of a system for such high velocities could be found.

The second order terms may in the future make themselves experimentally manifest through an increase of mass of rapidly moving a-particles.

9. Expression (18) may readily be verified for simple symmetrical systems. For a single charged conducting sphere of radius (a) the mass for slow velocities is well-known to be

${\frac {2}{3}}{\frac {1}{V^{2}}}{\frac {e^{2}}{a}}={\frac {2}{3}}{\frac {e}{V^{2}}}{\frac {e}{a}}={\frac {4}{3V^{2}}}{\frac {1}{2}}$ (e. Potential) $={\frac {4}{3V^{2}}}W$.

An interesting verification of equation (16) for the special case of a general, rigid, electrostatic system in translatory motion has been furnished me privately by Mr. G. F. C. Searle. He obtains for such a system (Phil. Mag. Jan. 1907, p. 129) the expression

$M_{x}={\frac {2T}{v_{1}}}$,

(19)

where $M_{x}$ is the momentum of the entire system along the direction ($x$) of motion, ($T$) is the total magnetic energy due to this motion, and ($c_{1}$) is the common translatory velocity possessed by all parts of the system.

Now it is well known that where the Faraday tubes move through space uniformly, as in the present case, the magnetic force ($H$) is given in terms of the electric force ($E$) by the expression

$H={\frac {v_{1}}{V}}E\ \sin \theta$,

($\theta$) representing the angle between ($E$) and the velocity of motion ($v_{1}$), and ($H$) being in a direction perpendicular both to ($E$) and ($v_{1}$). In the present notation

$H={\frac {v_{1}}{V}}{\sqrt {E_{y}^{2}+E_{z}^{2}}}$,

and hence we have

$T={\frac {H^{2}}{8\pi }}={\frac {1}{8\pi }}{\frac {v_{1}^{2}}{V^{2}}}(E_{y}^{2}+E_{z}^{2})$.

(20)