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

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 iii 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