# Page:LorentzGravitation1916.djvu/17

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

is a homogeneous linear function of ${\displaystyle X_{1},\dots X_{4}}$. 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 ${\displaystyle x_{a}}$. We can represent it by an expression of the form

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

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

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

vanish.

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

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

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

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

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

1. From § 10 it follows that if the length of a vector ${\displaystyle \mathrm {A} }$ 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 ${\displaystyle \mathrm {A} }$ 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.