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 X1, Y1, Z1, for which β and γ disappear in equations (2), α is equal to ϰ, that is, a co-ordinate system whose X1-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 δ1, δ2, δ3 and δ, against the system X1, Y1, Z1.
By these introductions our equations (3), (4) and (5) will be:
According to (4'), the three directions δ1, δ2, δ3 are perpendicular to each other, according to (5') falls into δ, then it must be: