Page:LorentzGravitation1916.djvu/17

 $\left[\mathrm{R}_{e}\cdot\mathrm{N}\right]_{x}$

is a homogeneous linear function of $$X_{1},\dots X_{4}$$. Under the special assumptions specified at the beginning of this § these are every where, the same functions. Let us thus consider a definite component of (15) e.g. that which corresponds to the direction of the coordinate $$x_{a}$$. We can represent it by an expression of the form

$\int\left(\alpha_{1}X_{1}+\dots+\alpha_{4}X_{4}\right)d\sigma$

where $$\alpha_{1},\dots\alpha_{4}$$ are constants. It will therefore be sufficient to prove that the four integrals

vanish.

In order to calculate $$\int X_{1}d\sigma$$ we consider an infinitely small prism, the edges of which have the direction $$x_1$$. This prism cuts from the boundary surface $$\sigma$$ two elements $$d\sigma$$ and $$\overline{d\sigma}$$. Proceeding along a generating line in the direction of the positive $$x_{1}$$ we shall enter the extension $$\Omega$$ bounded by $$\sigma$$ through one of these elements and leave it through the other. Now the vectors perpendicular to $$\sigma$$, which occur in (15) and which we shall denote by $$\mathrm{N}$$ and $$\bar{\mathrm{N}}$$ for the two elements, have the same value. If, therefore, $$S$$ and $$\bar{S}$$ are the absolute values of the projections of $$\mathrm{N}$$ and $$\bar{\mathrm{N}}$$ on a line in the direction $$x_1$$, we have according to (14)

Let first the four directions of coordinates be perpendicular to one another. Then the components of the vector obtained by projecting $$\mathrm{N}$$ on the above mentioned line are $$X_{1},0,0,0$$ and similarly those of the projection of $$\bar{\mathrm{N}}:\bar{X}_{1},0,0,0$$. But as, proceeding in the direction of $$x_1$$ we enter $$\Omega$$ through one element and leave it through the other, while $$\mathrm{N}$$ and $$\bar{\mathrm{N}}$$ are both directed outward, $$X_{1}$$ and $$\overline{X_{1}}$$, must have opposite signs. So we have

$S:\bar{S}=X_{1}:-\bar{X}_{1}$

and because of (17) we may now conclude that the elements $$X_{1}d\sigma$$