Page:LorentzGravitation1916.djvu/10

a vector may be resolved into four components which have the directions of the coordinates, viz. such directions that a shift along the first e.g. changes $x_{1}$, while $x_{2},x_{3},x_{4}$ remain constant. The four components in question are determined by the differentials $dx_{1},\dots dx_{4}$ corresponding to $PQ$. We shall say that by these they are expressed in "$x$-measure". Their values in natural measure are found by multiplying $dx_{1},\dots dx_{4}$ by certain factors. If we keep in mind that the radius-vectors of the e conjugate indicatrix and the indicatrix in the directions of the axes are expressed in "$x$ measure" by

${\frac {\epsilon }{\sqrt {-g_{11}}}},\ {\frac {\epsilon }{\sqrt {-g_{22}}}},\ {\frac {\epsilon }{\sqrt {-g_{33}}}},\ {\frac {\epsilon }{\sqrt {g_{44}}}},$

and in natural units by

$i\epsilon ,\ i\epsilon ,\ i\epsilon ,\ \epsilon$

we find for the reducing factors

$l_{1}=i{\sqrt {-g_{11}}},\ l_{2}=i{\sqrt {-g_{22}}},\ l_{3}=i{\sqrt {-g_{33}}},\ l_{4}=i{\sqrt {g_{44}}}.$

(7)

In the language of vector-analysis the vector obtained by the composition of two or more vectors is also called the sum of these vectors.

We shall also speak of finite vectors, i.e. of directed quantities which can be represented on an infinitely reduced scale by line-elements in the field-figure. If $\omega$ is the constant "reduction factor" chosen for this purpose, a vector $\mathrm {A}$ will be represented by a line-element $\omega \mathrm {A}$, the direction of which is also ascribed to $\omega \mathrm {A}$. It will now be evident that two finite vectors, as well as two infinitely small ones, determine an infinitesimal two dimensional extension and that finite vectors can be compounded and resolved by means of parallelograms and parallelepipeds. Also that we may speak of the "magnitude" of such figures, that e.g. the rule given in § 8 applies to the parallelogram described on two vectors.

The components of a vector in the directions of the coordinates expressed in $x$-measure will be called $X_{1},X_{2},X_{3},X_{4}$. This means that $\omega X_{1},\dots \omega X_{4}$ are equal to the differentials $dx_{1},\dots dx_{4}$ corresponding to the infinitely small vector $\omega \mathrm {A}$.

If we want to know the components of $\mathrm {A}$ in natural units we must multiply $X_{1},\dots X_{4}$ by the factors (7).

§ 11. Two vectors $\mathrm {A}$ and $\mathrm {B}$ starting from a point $P$ of the field-figure and lying in a plane $V$, determine what we shall call a rotation $\mathrm {R}$ in that plane. We ascribe to it the direction indicated by the order $\mathrm {AB}$ and a value given by the parallelogram described on