Page:Grundgleichungen (Minkowski).djvu/9

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(6) e'_{x'}=\frac{\mathfrak{e}_{x}-q\mathfrak{m}_{y}}{\sqrt{1-q^{2}}},\ \mathfrak{m}'_{y'}=\frac{-q\mathfrak{e}_{x}+\mathfrak{m}_{y}}{\sqrt{1-q^{2}}},\ \mathfrak{e}'_{z'}=\mathfrak{e}_{z}

and

(7) \mathfrak{m}'_{x'}=\frac{\mathfrak{m}_{x}+q\mathfrak{e}_{y}}{\sqrt{1-q^{2}}},\ \mathfrak{e}'_{y'}=\frac{q\mathfrak{m}_{x}+\mathfrak{e}_{y}}{\sqrt{1-q^{2}}},\ \mathfrak{m}'_{z'}=\mathfrak{m}_{z}[1]

Then we have for these newly introduced vectors \mathfrak{w',e',m'} with components \mathfrak{w}'_{x},\mathfrak{w}'_{y},\mathfrak{w}'_{z}; \mathfrak{e}'_{x},\mathfrak{e}'_{y},\mathfrak{e}'_{z}; \mathfrak{m}'_{x},\mathfrak{m}'_{y},\mathfrak{m}'_{z} and the quantity \varrho' a series of equations I'), II'), III'), IV) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here that e\mathfrak{e}_{x}-q\mathfrak{m}_{y},\ \mathfrak{e}_{y}+q\mathfrak{m}_{x},\ \mathfrak{e}_{z} are components of the vector \mathfrak{e}+[\mathfrak{vm}], where \mathfrak{v} is a vector in the direction of the positive z-axis, and \left|\mathfrak{v}\right|=q, and [\mathfrak{vm}] is the vector product of \mathfrak{v} and \mathfrak{m}; similarly \mathfrak{m}_{x}+q\mathfrak{e}_{y},\ \mathfrak{m}_{y}-q\mathfrak{e}_{x},\ \mathfrak{m}_{z} are the components of the vector \mathfrak{m}-[\mathfrak{ve}].

The equations 6) and 7), as they stand in pairs, can be expressed as.

\mathfrak{e}'_{x'}+i\mathfrak{m}'_{x'}=(\mathfrak{e}_{x}+i\mathfrak{m}_{x})\cos\ i\psi+(\mathfrak{e}_{y}+i\mathfrak{m}_{y})\sin\ i\psi,

\mathfrak{e}'_{y'}+i\mathfrak{m}'_{y'}=-(\mathfrak{e}_{x}+i\mathfrak{m}_{x})\sin\ i\psi+(\mathfrak{e}_{y}+i\mathfrak{m}_{y})\cos\ i\psi,

\mathfrak{e}'_{z'}+i\mathfrak{m}'_{z'}=\mathfrak{e}_{z}+i\mathfrak{m}_{z}

If \varphi denotes any other real angle, we can form the following combinations : —

(8) (\mathfrak{e'}_{x'}+i\mathfrak{m}'_{x'})\cos\ \varphi+(\mathfrak{e'}_{y'}+i\mathfrak{m}'_{y'})\sin\ \psi

=(\mathfrak{e}_{x}+i\mathfrak{m}_{x})\cos\ (\varphi+i\psi)+(\mathfrak{e}_{y}+i\mathfrak{m}_{y})\sin\ (\varphi+i\psi),

(9) -(\mathfrak{e'}_{x'}+i\mathfrak{m}'_{x'})\sin\ \varphi+(\mathfrak{e'}_{y'}+i\mathfrak{m}'_{y'})\cos\ \varphi

=-(\mathfrak{e}_{x}+i\mathfrak{m}_{x})\sin\ (\varphi+i\psi)+(\mathfrak{e}_{y}+i\mathfrak{m}_{y})\cos\ (\varphi+i\psi)

§ 4. Special Lorentz-Transformation.

The role which is played by the z-axis in the transformation (4)

  1. The equations (5) are written in a different order, however, equations (6) and (7) in the same order as the equations mentioned before, which amounts to them