Page:Grundgleichungen (Minkowski).djvu/9

(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