Page:LorentzGravitation1916.djvu/14

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must also be deferred. I shall conclude now by remarking that, as an immediate consequence of Hamilton's principle, the world-line of a material point which is acted on only by a given gravitation field, will be a geodetic line, and that the equations which determine the gravitation field caused by material and electromagnetic systems will be found by the consideration of infinitely small variations of the indicatrices, by which the numerical values of all quantities that are measured by means of these surfaces will be changed.

II.

(Communicated in the meeting of March 25, 1916).


§ 15. In the first part of this communication the connexion between the electric and the magnetic force on one hand and the charge and the convection current on the other was expressed by the equation

\int\left\{ \left[\mathrm{R}_{e}\cdot\mathrm{N}\right]+\left[\mathrm{R}_{h}\cdot\mathrm{N}\right]\right\} _{x}d\sigma=i\int\{\mathrm{q}\}_{x}d\Omega (10)

which has been discussed in § 13. It will now be shown that this formula is equivalent to the differential equations by which the connexion in question is expressed in the theory of Einstein. For this purpose some further geometrical considerations must first be developed. They refer to the special case that the quantities g_{ab} , have the same values at every point of the field-figure.

If this condition is fulfilled, considerations which generally may be applied to infinitesimal extensions only are valid for finite extensions too.


§ 16. The factor required, in the measurement of four-dimensional domains, for the passage from x-units to natural units has now the same value at every point of the field-figure. Similarly, when any one-, two- or three-dimensional extension in the field-figure that is determined by linear equations ("linear extensions") is considered, the factor by means of which the said passage may be effected for parts of that extension, will be the same for all those parts. Moreover the factor in question will be the same for two "parallel" extensions of this kind, i.e. for two extensions the determining equations of which can be written in such a way that the coefficients of x_{1},\dots x_{4} are the same in them.