(14) | , |

*further*^{[1]}

(15) | , |

*Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.*

The solution of the equations (10), (11), (12) leads to

(16) | . |

Now we shall make a very important observation about the vectors and . We can again introduce the indices 1, 2, 3, 4, so that we write instead of *x,' y,' z,' it'* , and instead of . Like the rotation round the *z*-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also *linear transformations* with the determinant +1, *so that*

(17) | d. i. |

*is transformed into*

On the basis of the equations (13), (14), we shall have

transformed into or in other words,

(18) | , |

is an *invariant* in a Lorentz-transformation.

- ↑ The brackets shall only summarize the expressions, which are related to the index, and shall denote the vector product of and .