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

142 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