Page:The Meaning of Relativity - Albert Einstein (1922).djvu/22

by the property that in them the measurable distance between two points, $s$, is expressed by the equation

$s^{2}=\sum \Delta x_{\nu }^{2}.$

If $K_{(x_{\nu })}$ and $K_{(x_{\nu })}'$ are two Cartesian systems of co-ordinates, then

$\sum \Delta x_{\nu }^{2}=\sum \Delta x_{\nu }'^{2}.$

The right-hand side is identically equal to the left-hand side on account of the equations of the linear orthogonal transformation, and the right-hand side differs from the left-hand side only in that the $x_{\nu }$ are replaced by the $x_{\nu }'$. This is expressed by the statement that $\sum \Delta x_{\nu }^{2}$ is an invariant with respect to linear orthogonal transformations. It is evident that in the Euclidean geometry only such, and all such, quantities have an objective significance, independent of the particular choice of the Cartesian co-ordinates, as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariants, which has to do with the laws that govern the form of invariants, is so important for analytical geometry.

As a second example of a geometrical invariant, consider a volume. This is expressed by

$V=\iiint dx_{1}dx_{2}dx_{3}.$

By means of Jacobi's theorem we may write

$\iiint dx_{1}'dx_{2}'dx_{3}'=\iiint {\frac {\delta (x_{1}',x_{2}',x_{3}')}{\delta (x_{1},x_{2},x_{3})}}dx_{1}dx_{2}dx_{3}$