Page:The Meaning of Relativity - Albert Einstein (1922).djvu/50

38 has a significance which is independent of the choice of the inertial system; but the invariance of the quantity $$\sum\Delta x_\nu^2$$ 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 $$\lambda$$, independent of $$v$$. 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 $$R_0$$, its radius $$R$$ in motion, might be different from $$R_0$$ 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 all directions in space must be equivalent to each other. $$R$$ may therefore depend upon the magnitude $$q$$ of the velocity, but not upon its direction; $$R$$ must therefore be an even function of $$q$$. If the cylinder is at rest relatively to $$K'$$ 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 $$K$$ satisfies the equation