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velocity v and β is defined as the ratio of v to the velocity of light estimated in the manner indicated in the first part of § 7, then to an observer on $S_{1}$ the unit of length of $S_{1}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{3}$ the unit of length of $S_{2}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (MVLA).

Corollary. Theorem V. may be stated as depending on (MVLR') instead of on {MVLA).

Theorem VI. If two systems of reference $S_{1}$ and $S_{2}$ move with a relative velocity v and if β is the ratio of v to the velocity of light, then to an observer on $S_{1}$ the unit of length of $S_{1}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{2}$ while to an observer on $S_{2}$ the unit of length of $S_{2}$ along the line of relative motion appears to be in the ratio ${\sqrt {1-\beta ^{2}}}:1$ to that of $S_{1}$ (MVLR).

We might make an analysis of these results similar to that which we gave for the corresponding results in the preceding section. But it would be largely a repetition. It is sufficient to point out that the remarkable conclusions as to units of length in the two systems rest on just those assumptions which led to the strange results as to the units of time.

§ 9. Simultaneity of Events Happening at Different Places. — Let us now assume two systems of reference S and $S'$ moving with a uniform relative velocity v. Let an observer on $S'$ undertake to adjust two clocks at different places so that they shall simultaneously indicate the same time. We will suppose that he does this in the following very natural manner:^[1] Two stations A and B are chosen in the line of relative motion of S and $S'$ and at a distance d apart. The point C midway between these two stations is found by measurement. The observer is himself stationed at C and has assistants at A and B. A single light signal is flashed from C to A and to B, and as soon as the light ray reaches each station the clock there is set at an hour agreed upon beforehand. The observer on $S'$ now concludes that his two clocks, the one at A and the other at B, are simultaneously marking the same hour; for, in his opinion (since he supposes his system to be at rest), the light has taken exactly the same time to travel from C to A as to travel from C to B.

Now let us suppose that an observer on the system S has watched the work of regulating these clocks on $S'$. The distances CA and CB appear to him to be

1. Compare Comstock, Science, N. S.. 31 (1900): 767-772.