construction, and which are considered at rest with respect to the
rotating reference-body, go at rates which are dependent on the
positions of the clocks. We shall now examine this dependence
quantitatively. A clock, which is situated at a distance r from the
centre of the disc, has a velocity relative to _K_ which is given by
_v_ = _r_,
where represents the angular velocity of rotation of the disc _K _
with respect to _K_. If _v_0, represents the number of ticks of the
clock per unit time ( rate of the clock) relative to _K_ when the
clock is at rest, then the rate of the clock (_v_) when it is moving
relative to _K_ with a velocity _v_, but at rest with respect to the
disc, will, in accordance with Section XII, be given by
image055
or with sufficient accuracy by
image056
This expression may also be stated in the following form:
image057
If we represent the difference of potential of the centrifugal force
between the position of the clock and the centre of the disc by ,
_i.e._ the work, considered negatively, which must be performed on the
unit of mass against the centrifugal force in order to transport it
from the position of the clock on the rotating disc to the centre of
the disc, then we have
image058
From this it follows that
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