Page:The Meaning of Relativity - Albert Einstein (1922).djvu/96

84 We have, first, by (67),

In this, $$\Gamma_{\alpha\beta}^\mu$$ is the value of this quantity at the variable point $$G$$ of the path of integration. If we put

and denote the value of $$\Gamma_{\alpha\beta}^\mu$$ at $$P$$ by $$\overline{\Gamma_{\alpha\beta}^\mu}$$, then we have, with sufficient accuracy,

Let, further, $$A^\alpha$$ be the value obtained from $$\overline{A^\alpha}$$ by a parallel displacement along the curve from $$P$$ to $$G$$. It may now easily be proved by means of (67) that $$A^\mu - \overline{A^\mu}$$ is infinitely small of the first order, while, for a curve of infinitely small dimensions of the first order, $$\Delta A^\mu$$ is infinitely small of the second order. Therefore there is an error of only the second order if we put

If we introduce these values of $$\Gamma_{\alpha\beta}^\mu$$ and $$A^\alpha$$ into the integral, we obtain, neglecting all quantities of a higher order of small quantities than the second,

The quantity removed from under the sign of integration