Page:Eddington A. Space Time and Gravitation. 1920.djvu/159

IX] construction made with the new velocities must again bring us to $$C$$; that is to say, the new velocities are represented by the directions $$OB^\prime$$, $$B^\prime C$$, where $$B^\prime$$ is some other point on the line $$NB$$.

Now examine how this will appear to some other observer $$S_1$$ in uniform motion relative to $$S$$. His transformation of space and time has been described in Chapter  and is represented in Fig. 20, which shows how his time-partitions run as compared with those of $$S$$. The same actual motion is, of course, represented by parallel directions in the two diagrams; but the interpretation as a velocity $$MA$$ is different in the two cases. Carrying the velocity of $$m_1$$ through two time-partitions, and of $$m_2$$ through three time-partitions, as before, we find that the total momentum for the observer $$S_1$$ is represented by $$PC$$ (Fig. 20); but making a similar construction with the velocities after collision, we arrive at a different point $$C^\prime$$. Thus whilst momentum is conserved for the observer $$S$$, it has altered from $$PC$$ to $$PC^\prime$$ for the observer $$S_1$$.

The discrepancy arises because in the construction the lines are prolonged to meet partitions which are different for the two