# 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