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which we are considering is the same in both systems, the observer A, always using the units of his own system, concludes that the change in velocity of the ball b is smaller in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$ than the change in velocity of the ball a. The change in velocity of each ball multiplied by its mass gives its change in momentum. Now, from the law of conservation of momentum, A assumes that each ball experiences the same change in momentum, and therefore since he has already decided that the ball b has experienced a smaller change of velocity in the ratio ${\frac {\sqrt {1-\beta ^{2}}}{1}}$, he must conclude that the mass of the ball in system b is greater than that of his own in the ratio ${\frac {1}{\sqrt {1-\beta ^{2}}}}$.

In general, therefore, we must assume that the mass of a body increases with its velocity. We must bear in mind, however, as in all other cases, that the motion is determined with respect to some point arbitrarily chosen as a point of rest.

If m is the mass of a body in motion, and m₀ its mass at rest, we have^[1]

${\frac {m}{m_{0}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}$.

I

The only opportunity of testing experimentally the change of a body's mass with its velocity has been afforded by the experiments on the mass of a moving electron, or β particle. The actual measurements were indeed not of the mass of the electron, but of the ratio of charge to mass $\left({\frac {e}{m}}\right)$. It has, however, been universally considered that the charge e is constant. In other words, that the force acting upon the electron in a uniform electrostatic field is independent of its velocity relative to the field. Hence the observed change in ${\frac {e}{m}}$ is attributed solely to the change in mass. It might be well to subject this view to a more careful analysis than has hitherto been done. At present, however, we will adopt it without further scrutiny.

The original experiments of Kaufmann^[2] showed only a qualitative

1. This equation and others developed in this section are identical with those obtained through an entirely different course of reasoning by Lewis (Phil. Mag., 16, 705 (1908)). The equations were there obtained for systems in motion with respect to a point at absolute rest. We shall show here, however, that they are true, whatever arbitrary point is selected as a point of rest.
2. See Lewis, loc. cit.