Page:Relativity (1931).djvu/176

tion, 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

$\nu =\omega r$,

where $\omega$ 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 ($\nu$) when it is moving relative to $K$ with a velocity $\nu$, but at rest with respect to the disc, will, in accordance with Section XII, be given by

$\nu =\nu _{0}{\sqrt {1-{\frac {\nu ^{2}}{c^{2}}}}}$

or with sufficient accuracy by

$\nu =\nu _{0}\left(1-{\frac {1}{2}}{\frac {\nu ^{2}}{c^{2}}}\right)$.

This expression may also be stated in the following form:

$\nu =\nu _{0}\left(1-{\frac {1}{c^{2}}}{\frac {\omega ^{2}r^{2}}{2}}\right)$.

If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by $\phi$, i.e. the work,