Page:Grundgleichungen (Minkowski).djvu/10

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can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law: —

Let be a vector with the components , and let . By we shall denote any vector which is perpendicular to , and by , we shall denote components of in direction of and .

Instead of x, y, z, t, new magnetudes x,' y,' z,' t' will be introduced in the following way. If for the sake of shortness, is written for the vector with the components x, y, z in the first system of reference, for the same vector with the components x', y', z' in the second system of reference, then for the direction of we have

(10) ,

and for every perpendicular direction

(11) ,

and further

(12)

The notations and are to be understood in the sense that with the directions , and every direction perpendicular to in the system x, y, z are always associated the directions with the same direction cosines in the system x', y', z' ,

A transformation which is accomplished by means of (10), (11), (12) with the condition will be called a special Lorentz-transformation. We shall call the vector, the direction of the axis, and the magnitude of the moment of this transformation.

If further and the vectors , in the system x', y', z' are so defined that,

(13) ,