Let us now consider a seconds-clock which is permanently situated at
the origin (_x _ = 0) of _K _. _t _ = 0 and _t _ = 1 are two successive
ticks of this clock. The first and fourth equations of the Lorentz
transformation give for these two ticks:
_t_ = 0
and
image011
As judged from _K_, the clock is moving with the velocity _v_; as
judged from this reference-body, the time which elapses between two
strokes of the clock is not one second, but
image012
seconds, _i.e._ a somewhat larger time. As a consequence of its motion
the clock goes more slowly than when at rest. Here also the velocity
_c_ plays the part of an unattainable limiting velocity.
XIII.
THEOREM OF THE ADDITION OF VELOCITIES. THE EXPERIMENT OF FIZEAU
Now in practice we can move clocks and measuring-rods only with
velocities that are small compared with the velocity of light; hence we
shall hardly be able to compare the results of the previous section
directly with the reality. But, on the other hand, these results must
strike you as being very singular, and for that reason I shall now draw
another conclusion from the theory, one which can easily be derived
from the foregoing considerations, and which has been most elegantly
confirmed by experiment.
In Section VI we derived the theorem of the addition of velocities in
one direction in the form which also results from the hypotheses of
classical mechanics. This theorem can also be deduced readily from the
28