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${\frac {1}{2}}d{\sqrt {1-\beta ^{2}}}$

instead of ½d. Moreover, since the velocity of light is independent of the velocity of the source, it appears to him that the light ray proceeding from C to A has approached A at the velocity c + v, where c is the velocity of light, while the light ray going from C to B has approached B at the velocity c — v. Thus to him it appears that the light has taken longer to go from C to B than from C to A by the amount

${\frac {{\frac {1}{2}}d{\sqrt {1-\beta ^{2}}}}{c-v}}-{\frac {{\frac {1}{2}}d{\sqrt {1-\beta ^{2}}}}{c+v}}={\frac {vd{\sqrt {1-\beta ^{2}}}}{c^{2}-v^{2}}}$

But since $\beta =v/c$ the last expression is readily found to be equal to

${\frac {v}{c^{2}}}\cdot {\frac {d}{\sqrt {1-\beta ^{2}}}}$.

Therefore, to an observer on S the clocks on $S'$ appear to mark different times; and the difference is that given by the last expression above.

Thus we have the following conclusion:

Theorem VII. Let two systems of reference S and S' have a uniform relative velocity v. Let an observer on S' place two clocks at a distance d apart in the line of relative motion of S and S' and adjust them so that they appear to him to mark simultaneously the same time. Then to an observer on S the clock on S' which is forward in point of motion appears to be behind in point of time by the amount

${\frac {v}{c^{2}}}\cdot {\frac {d}{\sqrt {1-\beta ^{2}}}}$,

where c is the velocity of light and $\beta =v/c$ (MVLR).

It should be emphasized that the clocks on $S'$ are in agreement in the only sense in which they can be in agreement for an observer on that system who supposes (as he naturally will) that his own system is at rest — notwithstanding the fact that to an observer on the other system there appears to be an irreconcilable disagreement depending for its amount directly on the distance apart of the two clocks.

According to the result of the last theorem the notion of simultaneity of events happening at different places is indefinite in meaning until some convention is adopted as to how simultaneity is to be determined. In other words, there is no such thing as the absolute simultaneity of events happening at different places.

§ 10. Transformation of Space and Time Coordinates. — It is now an easy matter to derive the Einstein formulae^[1] for the transformation of space

1. Jahrbuch der Radioaktivität, 4 (1907): 418-420.