Page:LorentzGravitation1916.djvu/9

(1,2) (2,3), (3,1), which do not intersect the indicatrix itself but the conjugate indicatrix, this proposition follows from a well-known theorem of Gauss in the theory of curvature of surfaces; for the other three (1,4), (2,4), (3,4), which cut the indicatrix itself, the proof can be given by direct calculation. The considerations necessary for this, and some other calculations with which we shall be concerned further on will be communicated in a later paper.

In considering the three last-mentioned extensions I have confined myself to triangles with real sides (§ 7, b).

The quotient

${\frac {e}{\Delta }}=K_{ab}$

is now for each extension a definite number, which we may consider as a measure of the curvature of the two-dimensional extension ($a,b$); the sum $K$ of the six numbers $K_{ab}$ may be called the curvature of the field-figure at the point $P$ in question. This quantity is the same that has been introduced by Hilbert; this results from the calculation of its value, which at the same time shows $K$ to be independent of the special choice of the directions 1, 2, 3, 4 introduced in the beginning of this §.

The numbers $K_{ab}$ all real and have a meaning that can be indicated without the introduction of coordinates; moreover their sum $K$ is not changed by a deformation of the field-figure.

If now $d\Omega$ is an element of the four-dimensional extension of the field-figure, expressed in natural measure, the part of the principal function belonging to the gravitation field is

$H_{3}={\frac {i}{\varkappa }}\int Kd\Omega$

(6)

where the integration is extended to the domain considered (§ 6) while $\varkappa$ is the gravitation constant. $H_{3}$ too is not changed by a deformation of the field-figure.

The factor $i$ has been introduced in order to obtain a real value for $H_{3}$, the element $d\Omega$ being represented in natural measure by a negative imaginary number (§ 8).

§ 10. What we have to say of the electromagnetic field must be preceded by some considerations belonging to what may be called the "vector theory" of the field-figure.

A line-element $PQ$, taken in a definite, direction (indicated by the order of the letters), may be called a vector. Such vectors can be compounded or decomposed by means of parallelograms or parallelepipeds. Especially, when coordinates $x_{1},\dots x_{4}$ have been chosen,