the shortening is the larger, the larger is _v_. For _v_ = _c_, all moving bodies, when considered from a stationary system shrink into planes. For a velocity larger than the velocity of light, our propositions become meaningless; in our theory _c_ plays the part of infinite velocity. It is clear that similar results hold about stationary bodies in a stationary system when considered from a uniformly moving system. Let us now consider that a clock which is lying at rest in the stationary system gives the time _t_, and lying at rest relative to the moving system is capable of giving the time τ; suppose it to be placed at the origin of the moving system _k_, and to be so arranged that it gives the time τ. How much does the clock gain, when viewed from the stationary system K? We have, $$ \tau = \frac {1}{\sqrt {1-\frac {v^2}{c^2}}} (t - \frac {v}{c^2}x), $$ and _x_ = _vt_, $$ \therefore \tau - t = (1 - \sqrt {1 - \frac {v^2}{c^2}}) t. $$ Therefore the clock loses by an amount ½(_v²_/_c²_) per second of motion, to the second order of approximation. From this, the following peculiar consequence follows. Suppose at two points A and B of the stationary system two clocks are given which are synchronous in the sense explained in § 3 when viewed from the stationary system. Suppose the clock at A to be set in motion in the line joining it with B, then after the arrival of the clock at B, they will no longer be found synchronous, but the clock which was set in motion from A will lag behind the clock which had been all along at B by an amount ½_t_(_v²_/_c²_), where _t_ is the time required for the journey. 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 ½_t_(_v²_/_c²_) 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__{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 $$ x = (\frac {w_{\xi} + v}{1+\frac {vw_{\xi}}{c^2}}) t $$,