Page:LorentzGravitation1916.djvu/17

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\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.