Page:Eddington A. Space Time and Gravitation. 1920.djvu/158

It was formerly supposed that the mass of a particle was a number attached to the particle, expressing an intrinsic property, which remained unaltered in all its vicissitudes. If $m$ is this number, and $u$ the velocity of the particle, the momentum is $mu$; and it is through this relation, coupled with the law of conservation of momentum that the mass $m$ was defined. Let us take for example two particles of masses $m_{1}=2$ and $m_{2}=3$, moving in the same straight line. In the space-time diagram for an observer $S$ the velocity of the first particle will be represented by a direction $OA$ (Fig. 19). The first particle moves through

a space $MA$ in unit time, so that $MA$ is equal to its velocity referred to the observer $S$. Prolonging the line $OA$ to meet the second time-partition, $NB$ is equal to the velocity multiplied by the mass 2; thus the horizontal distance $NB$ represents the momentum. Similarly, starting from $B$ and drawing $BC$ in the direction of the velocity of $m_{2}$, prolonged through three time-partitions, the horizontal progress from $B$ represents the momentum of the second particle. The length $PC$ then represents the total momentum of the system of two particles.

Suppose that some change of their velocities occurs, not involving any transference of momentum from outside, e.g. a collision. Since the total momentum $PC$ is unaltered, a similar