Page:Eddington A. Space Time and Gravitation. 1920.djvu/161

invariant mass multiplied by a modified velocity $\partial x/\partial s$. The physicist, however, prefers for practical purposes to keep to the old definition of momentum as mass multiplied by the velocity $\partial x/\partial t$. We have $m{\frac {\partial x}{\partial s}}=m{\frac {\partial t}{\partial s}}\cdot {\frac {\partial x}{\partial t}}$, accordingly the momentum is separated into two factors, the velocity $\partial x/\partial t$, and a mass $M=m\partial t/\partial s$, which is no longer an invariant for the particle but depends on its motion relative to the observer's space and time. In accordance with the usual practice of physicists the mass (unless otherwise qualified) is taken to mean the quantity $M$.

Using unaccelerated rectangular axes, we have by definition of $s$ $\partial s^{2}=\partial t^{2}-\partial x^{2}-\partial y^{2}-\partial x^{2}$, so that ${\begin{aligned}\left({\frac {\partial s}{\partial t}}\right)^{2}&=1-\left({\frac {\partial x}{\partial t}}\right)^{2}-\left({\frac {\partial y}{\partial t}}\right)^{2}-\left({\frac {\partial z}{\partial t}}\right)^{2}{\text{,}}\\&=1-u^{2}{\text{,}}\end{aligned}}$ where $u$ is the resultant velocity of the particle (the velocity of light being unity). Hence $M={\frac {m}{\sqrt {1-u^{2}}}}$. Thus the mass increases as the velocity increases, the factor being the same as that which determines the FitzGerald contraction.

The increase of mass with velocity is a property which challenges experimental test. For success it is necessary to be able to experiment with high velocities and to apply a known force large enough to produce appreciable deflection in the fast-moving particle. These conditions are conveniently fulfilled by the small negatively charged particles emitted by radioactive substances, known as β particles, or the similar particles which constitute cathode rays. They attain speeds up to 0.8 of the velocity of light, for which the increase of mass is in the ratio 1.66; and the negative charge enables a large electric or magnetic force to be applied. Modern experiments fully confirm the theoretical increase of mass, and show that the factor $1/{\sqrt {(1-u^{2})}}$