Page:Grundgleichungen (Minkowski).djvu/9



and

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.

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

§ 4. Special Lorentz-Transformation.
The role which is played by the z-axis in the transformation (4)