Page:The Meaning of Relativity - Albert Einstein (1922).djvu/112

100 the naturally measured lengths and times. Since $$ds^2$$, on the other hand, is known in terms of the co-ordinates $$x_\nu$$ employed in finite regions, in the form

we have the possibility of getting the relation between naturally measured lengths and times, on the one hand, and the corresponding differences of co-ordinates, on the other hand. As the division into space and time is in agreement with respect to the two systems of co-ordinates, so when we equate the two expressions for $$ds^2$$ we get two relations. If, by (101a), we put

we obtain, to a sufficiently close approximation, {{MathForm2|(106)| $$ \left. \begin{align} \sqrt{dX_1^2 + dX_2^2 + dX_3^2} & = \left(1 + \frac{\kappa}{8\pi}\int\frac{\sigma dV_0}{r}\right)\sqrt{dx_1^2 + dx_2^2 + dx_3^2} \\ dT & = \left(1 - \frac{\kappa}{8\pi}\int\frac{\sigma dV_0}{r}\right)dl. \end{align} \right\}. $$}}

The unit measuring rod has therefore the length,

in respect to the system of co-ordinates we have selected. The particular system of co-ordinates we have selected