These equations form the well-known special Lorentz transformation, which in the general theory represents a rotation, through an imaginary angle, of the four-dimensional system of co-ordinates. If we introduce the ordinary time , in place of the light-time , then in (29) we must replace by and by . We must now fill in a gap. From the principle of the constancy of the velocity of light it follows that the equation has a significance which is independent of the choice of the inertial system; but the invariance of the quantity does not at all follow from this. This quantity might be transformed with a factor. This depends upon the fact that the right-hand side of (29) might be multiplied by a factor , independent of . But the principle of relativity does not permit this factor to be different from 1, as we shall now show. Let us assume that we have a rigid circular cylinder moving in the direction of its axis. If its radius, measured at rest with a unit measuring rod is equal to , its radius in motion, might be different from , since the theory of relativity does not make the assumption that the shape of bodies with respect to a space of reference is independent of their motion relatively to this space of reference. But [Pg 36] all directions in space must be equivalent to each other. may therefore depend upon the magnitude of the velocity, but not upon its direction; must therefore be an even function of . If the cylinder is at rest relatively to ' the equation of its lateral surface is If we write the last two equations of (29) more generally then the lateral surface of the cylinder referred to satisfies the equation The factor therefore measures the lateral contraction of the cylinder, and can thus, from the above, be only an even function of . If we introduce a third system of co-ordinates, ", which moves relatively to ' with velocity in the direction of the negative -axis of , we obtain, by applying (29) twice, Now, since must be equal to and since we assume that we use the same measuring rods in all the systems, it follows that the transformation of " to must be the identical [Pg 37] transformation (since the possibility does not need to be considered). It is essential for these considerations to assume that the behaviour of the measuring rods does not depend upon the history of their previous motion. Moving Measuring Rods and Clocks. At the definite -time, , the position of the points given by the integers , is with respect to , given by ; this follows from the first of equations (29) and expresses the Lorentz contraction. A clock at rest at the origin of , whose beats are characterized by , will, when observed from ', have beats characterized by