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

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