Page:The Meaning of Relativity - Albert Einstein (1922).djvu/19

$\Delta x_{\nu }'=\sum \limits _{\alpha }{\frac {\delta x_{\nu }'}{\delta x_{\alpha }}}\Delta x_{\alpha }+{\frac {1}{2}}\sum \limits _{\alpha \beta }{\frac {\delta ^{2}x_{\nu }'}{\delta x_{\alpha }\delta x_{\beta }}}\Delta x_{\alpha }\Delta x_{\beta }$

If we substitute (2a) in this equation and compare with (1), we see that the $x_{\nu }'$, must be linear functions of the $x_{\nu }$. If we therefore put

$x_{\nu }'=a_{\nu }+\sum \limits _{\alpha }b_{\nu \alpha }x_{\alpha }$

(3)

${\text{or }}\Delta x_{\nu }'=\sum \limits _{\alpha }b_{\nu \alpha }\Delta x_{\alpha }$

(3a)

then the equivalence of equations (2) and (2a) is expressed in the form

$\sum \Delta x_{\nu }'^{2}=\lambda \sum \Delta x_{\nu }^{2}{\text{ (}}\lambda {\text{ independent of }}\Delta x_{\nu }{\text{)}}$

(2b)

It therefore follows that $\lambda$ must be a constant. If we put $\lambda =1$, (2b) and (3a) furnish the conditions

$\sum \limits _{\nu }b_{\nu \alpha }b_{\nu \beta }=\delta _{\alpha \beta }$

(4)

in which $\delta _{\alpha \beta }=1$, or $\delta _{\alpha \beta }=0$, according as $\alpha =\beta$ or $\alpha \not =\beta$. The conditions (4) are called the conditions of orthogonality, and the transformations (3), (4), linear orthogonal transformations. If we stipulate that $s^{2}=\sum \Delta x_{\nu }^{2}$ shall be equal to the square of the length in every system of co-ordinates, and if we always measure with the same unit scale, then $\lambda$ must be equal to 1. Therefore the linear orthogonal transformations are the only ones by means of which we can pass from one Cartesian system of co-ordinates in our space of reference to another. We see