XXXII

aroused by the war, these societies equipped two expeditions–-to Sobral (Brazil), and to the island of Principe (West Africa)–-and sent several of Britain's most celebrated astronomers (Eddington, Cottingham,

Crommelin, Davidson), in order to obtain photographs

of the solar eclipse of 29th May, 1919. The relative discrepancies to be expected between the stellar photographs obtained during the eclipse and the comparison photographs amounted to a few hundredths of a millimetre only. Thus great accuracy was necessary in making the adjustments required for the taking of the photographs, and in their subsequent measurement.

The results of the measurements confirmed the theory in a thoroughly satisfactory manner. The rectangular components of the observed and of the calculated

deviations of the stars (in seconds of arc) are set forth in the following table of results: ${@{}c*{2}{>{\quad}cc}@{}}

\ColHead{1}{Number of}{Number of\ the Star.} & \ColHead{2}{Observed. Calculated.}{First Co-ordinate. \$\overbrace{\text{Observed. Calculated.}}$\}\&\mathrm{\backslash ColHead}2Observed.Calculated.SecondCo-ordinate.$\overbrace{\text{Observed. Calculated.}}$$\begin{gathered} \} \\ 11\&-0.19\&-0.22\&+0.16\&+0.02 \\ \phantom{0}5\&+0.29\&+0.31\&-0.46\&-0.43 \\ \phantom{0}4\&+0.11\&+0.10\&+0.83\&+0.74 \\ \phantom{0}3\&+0.20\&+0.12\&+1.00\&+0.87 \\ \phantom{0}6\&+0.10\&+0.04\&+0.57\&+0.40 \\ 10\&-0.08\&+0.09\&+0.35\&+0.32 \\ \phantom{0}2\&+0.95\&+0.85\&-0.27\&-0.09 \end{gathered}$$

Displacement of Spectral Lines towards the Red

In [chapter:XXIII]Section XXIII it has been shown that in a system $K^{\prime }$ which is in rotation with regard to a Galileian system $K$, clocks of identical 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 $v=\omega r,$ where $\omega$ represents the angular velocity of rotation of the disc $K^{\prime }$ with respect to $K$. If ${\nu }_0$ 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 $v$, but at rest with respect to the disc, will, in accordance with [chapter:XII]Section XII, be given by $\nu ={\nu }_0\sqrt{1-\frac{v^2}{c^2}},$

or with sufficient accuracy by $\nu ={\nu }_0(1-\frac{1}{2}\frac{v^2}{c^2}).$ This expression may also be stated in the following form: $\nu ={\nu }_0(1-\frac{1}{c^2}\frac{{\omega }^2{r}^2}{2}).$ If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by $\varphi$, 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 $\varphi =-\frac{{\omega }^2{r}^2}{2}.$ From this it follows that $\nu ={\nu }_0(1+\frac{\varphi }{c^2}).$ In the first place, we see from this expression that two clocks of identical construction will go at different rates when situated at different distances from the centre of the disc. This result is also valid from the standpoint of an observer who is rotating with the disc.

Now, as judged from the disc, the latter is in a gravitational