Page:The Meaning of Relativity - Albert Einstein (1922).djvu/70

58 must hold. We shall now show that this equation leads to the same law of motion of a material particle as that already obtained. Let us imagine the matter to be of infinitely small extent in space, that is, a four-dimensional thread; then by integration over the whole thread with respect to the space co-ordinates $$x_1, x_2, x_3$$, we obtain

Now $$\int dx_1dx_2dx_3dx_4$$ is an invariant, as is, therefore, also $$\int\sigma_0dx_1dx_2dx_3dx_4$$. We shall calculate this integral, first with respect to the inertial system which we have chosen, and second, with respect to a system relatively to which the matter has the velocity zero. The integration is to be extended over a filament of the thread for which $$\sigma_0$$ may be regarded as constant over the whole section. If the space volumes of the filament referred to the two systems are $$dV$$ and $$dV_0$$ respectively, then we have

and therefore also

If we substitute the right-hand side for the left-hand side in the former integral, and put $$\frac{dx_1}{d\tau}$$ outside the sign