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

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

or with sufficient accuracy by

This expression may also be stated in the following form:

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,