Page:BumsteadContraction.djvu/8

In order for these periods to be equal we must have $\scriptstyle {m_{2}=(1-\beta ^{2})m_{1}}$ which is the same relation as that between the longitudinal and transverse masses of Lorentz's electron. That the variation with the velocity of $\scriptstyle {m_{1}}$ or $\scriptstyle {m_{2}}$ for ordinary matter is also the same as for Lorentz's electron may be shown in many ways; the following simple example will suffice for the purpose.

Consider an elastic rod with its length perpendicular to the motion of the earth and making longitudinal vibrations. If its period of vibration is $\scriptstyle {T}$ we shall have $\scriptstyle {T\propto {\sqrt {\frac {m_{2}}{\kappa }}}}$ where $\scriptstyle {m_{2}}$ is the transverse mass of any particle and $\scriptstyle {\kappa }$ is the coefficient of stretching of the rod. We must also have, by Einstein's transformation, $\scriptstyle {T={\frac {T_{0}}{\sqrt {1-\beta ^{2}}}}}$ where $\scriptstyle {T_{0}}$ is the period of the rod when at rest.^[1]

The constant $\scriptstyle {\kappa }$ depends on the intermolecular forces in the direction of the length of the rod, that is perpendicular to the earth's motion; and these must vary with the velocity in the same manner as electrical forces. If we have two point charges moving through the ether in a direction perpendicular to the line joining them, the force between them is $\scriptstyle {E=E_{0}{\sqrt {1-\beta ^{2}}}}$ where $\scriptstyle {E_{0}}$ is the force when they are at rest.^[2] Thus we have $\scriptstyle {\kappa =\kappa _{0}{\sqrt {1-\beta ^{2}}}}$ and $\scriptstyle {{\sqrt {\frac {m_{0}}{\kappa _{0}{\sqrt {1-\beta ^{2}}}}}}={\sqrt {\frac {m_{0}}{\kappa }}}{\frac {1}{\sqrt {1-\beta ^{2}}}}}$ whence $\scriptstyle {m_{2}=m_{0}{\frac {1}{\sqrt {1-\beta ^{2}}}}}$ It follows therefore from our hypothesis not only that all mass is electromagnetic but also that it varies with the speed in the specific manner of Lorentz's electron.

1. If this relation did not hold for any time-keeper, the velocity of light measured in a moving system would be different from that measured in a system at rest, and thus the principle of relativity would be violated.
2. See below, p. 503.