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 (1/2)t(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
ξ = w ?]_ξτ, η = w_ητ, ζ = 0,
where w_ξ and w_η 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 = ((w_ξ + v)/(1 + vw_ξ/c^2))t, y = (1 - v^2/c^2)^{1/2}w_ηt/(1 + vw_ξ/c^2), z = 0.
