# 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

 $\int X_{1}d\sigma\dots\int X_{4}d\sigma$ (16)

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.[1] 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)

 $Sd\sigma=\bar{S}\overline{d\sigma}$ (17)

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 $X_{1}d\sigma$

1. From § 10 it follows that if the length of a vector $\mathrm{A}$ that is represented by a line (§ 17) coincides with a radius-vector of the conjugate indicatrix, it is always represented by an imaginary number. We may however obtain a vector which in natural units is represented by a real number e.g. by 1 (§ 13) if we multiply the vector $\mathrm{A}$ by an imaginary factor, which means that its components and also those of a vector product in which it occurs are multiplied by that factor.