4) |

5) |

If we take as given, then we have 12 constants available, so we can arbitrarily use three of them.

The solution is most comfortable when we use a temporary co-ordinate system *X*_{1}, *Y*_{1}, *Z*_{1}, for which β and γ disappear in equations (2), α is equal to ϰ, that is, a co-ordinate system whose X_{1}-axis falls in the direction, of which the direction cosine is proportional to *X, Y, Z* with α, β, γ.

Furthermore, it should be set

then μ, ν, π are the direction cosines of 4 directions, which we will denote by δ_{1}, δ_{2}, δ_{3} and δ, against the system X_{1}, Y_{1}, Z_{1}.

By these introductions our equations (3), (4) and (5) will be:

3') |

4') |

5') |

According to (4'), the three directions δ_{1}, δ_{2}, δ_{3} are perpendicular to each other, according to (5') falls into δ, then it must be:

6) |