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

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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

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

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

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