Page:Grundgleichungen (Minkowski).djvu/12

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If we divide by this magnitude, we obtain the four values

,

so that

(19) .

It is apparent that these four values, are determined by the vector and inversely the vector of magnitude follows from the 4 values , where are real, real and positive and condition (19) is fulfilled.

The meaning of here is, that they are the ratios of to

(20)

The differentials denoting the displacements of matter occupying the spacetime point to the adjacent space-time point.

After the Lorentz-transfornation is accomplished the velocity of matter in the new system of reference for the same space-time point x', y', z', t' is the vector with the ratios as components.

Now it is quite apparent that the system of values

is transformed into the values

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocity after the transformation as the first system of values has got for before transformation.