Page:SahaElectrodynamics.djvu/16

We see forthwith that the result holds also when the clock moves from A to B by a polygonal line, and also when A and B coincide.

If we assume that the result obtained for a polygonal line holds also for a curved line, we obtain the following law. If at A, there be two synchronous clocks, and if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in t-seconds, then after arrival, the last mentioned clock will be behind the stationary one by ${\frac {1}{2}}t{\frac {v^{2}}{c^{2}}}$ seconds. From this, we conclude that a clock placed at the equator must be slower by a very small amount than a similarly constructed clock which is placed at the pole, all other conditions being identical.

§ 5. Addition-Theorem of Velocities.

Let a point move in the system k (which moves with velocity v along the x-axis of the system K) according to the equation

$\xi =w_{\xi }\tau ,\ \eta =w_{\eta }\tau ,\ \zeta =0$,

where $w_{\xi }$ and $w_{\eta }$ are constants.

It is required to find out the motion of the point relative to the system K. If we now introduce the system of equations in § 3 in the equation of motion of the point, we obtain

$x={\frac {w_{\xi }+v}{1+{\frac {vw_{\xi }}{c^{2}}}}},\ y={\frac {\left(1-{\frac {v^{2}}{c^{2}}}\right)^{\frac {1}{2}}w_{\eta }t}{1+{\frac {vw_{\xi }}{c^{2}}}}},\ z=0$.