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 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
where and 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