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postulates V, L, R, were still valid it would turn out that the two observers would find their time units in agreement. But, in view of M, the choice of S as fixed would undoubtedly seem perfectly arbitrary to both observers; and the content of the modified postulate R would be essentially different from that of the postulate as we have employed it. Hence, if we accept R as it stands — or, indeed, even a certain part of it, as we have shown above — we must conclude that the time units in the two systems are not in agreement, in fact, that their ratio is that stated in the theorems above.

§ 8. Relation Between the Units of Length. — Let us consider three systems of reference S, $S_{1}$ and $S_{2}$ related in the same manner as in the preceding section except that now the two lines $l_{1}$ and $l_{2}$ coincide. We suppose that $S_{1}$ is moving in the direction indicated by the arrow at A and that $S_{2}$ is moving in the direction indicated by the arrow at B.

We suppose that observers at A and B again measure the velocity of light as it emanates from S, this time in the direction of the line of motion. Each will carry out his observations on the supposition that his system is at rest, for from M it follows that he cannot detect the motion of his system. The observer at A measures the time $t_{1}$ of passage of a ray of light from A to C and return to A, the length of AC being d when the measurement is made with a unit belonging to $S_{1}$ . Likewise, the observer at B measures the time $t_{2}$ of passage of a ray of light from B to D and return to B, the length of BD being d when measured with a unit belonging to $S_{2}$ .

Just as in the preceding case it may be shown that the two observers must obtain the same estimate for the velocity of light. But the estimate of the observer at A is $2d/t_{1}$ while that of the observer at B is $2d/t_{2}$ . Hence

t_{1}=t_{2}

;

that is, the number of units of time required for the passage of the ray at A and of the ray at B is the same, the former being measured on $S_{1}$ and the latter on $S_{3}$ . Moreover, the measure of length is the same in the two cases. But the units of time, as we saw in the preceding section, do not have the same magnitude. Hence the units of length of the two systems along their line of motion do not have the same magnitude; and the ratio of units of length is the same as the ratio of units of time.

Combining this result with theorem III, its corollary, and theorem IV. we have the following three results:

Theorem V. If two systems of reference $S_{1}$ and $S_{2}$ move with a relative