${\frac {\beta \gamma }{\omega ^{2}}}=m_{2}m_{3}+n_{2}n_{3}+p_{2}p_{3}$
${\frac {\gamma \alpha }{\omega ^{2}}}=m_{3}m_{1}+n_{3}n_{1}+p_{3}p_{1}$
${\frac {\alpha \beta }{\omega ^{2}}}=m_{1}m_{2}+n_{1}n_{2}+p_{1}p_{2}$

4)

${\frac {\alpha }{\omega ^{2}}}=am_{1}+bn_{1}+cp_{1}$
${\frac {\beta }{\omega ^{2}}}=am_{2}+bn_{2}+cp_{2}$
${\frac {\gamma }{\omega ^{2}}}=am_{3}+bn_{3}+cp_{3}$

5)

If we take $\alpha \beta \gamma$ 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 coordinate system X_{1}, Y_{1}, Z_{1}, for which β and γ disappear in equations (2), α is equal to ϰ, that is, a coordinate 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
${\begin{array}{clcclcclcclc}m_{h}^{2}+n_{h}^{2}+p_{h}^{2}&=&q_{h}^{2},&m_{h}/q_{h}&=&\mu _{h},&n_{h}/q_{h}&=&\nu _{h},&p_{h}/q_{h}&=&\pi _{h}\\\\a^{2}+b^{2}+c^{2}&=&d^{2},&a/d&=&\mu ,&b/d&=&\nu ,&c/d&=&\pi ,\end{array}}$
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:
$1\omega ^{2}d^{2}=q_{1}^{2}{\frac {\varkappa ^{2}}{\omega ^{2}}}=q_{2}^{2}=q_{3}^{2}$

3')

$\mu _{2}\mu _{3}+\nu _{2}\nu _{3}+\pi _{2}\pi _{3}=\mu _{3}\mu _{1}+\nu _{3}\nu _{1}+\pi _{3}\pi _{1}=\mu _{1}\mu _{2}+\nu _{1}\nu _{2}+\pi _{1}\pi _{2}=0$
${\text{that is }}\cos(\delta _{2},\delta _{3})=\cos(\delta _{3},\delta _{1})=\cos(\delta _{1},\delta _{2})=0$

4')

$\mu \mu _{1}+\nu \nu _{1}+\pi \pi _{1}={\frac {\varkappa }{\omega ^{2}q_{1}d}},\ \mu \mu _{2}+\nu \nu _{2}+\pi \pi _{2}+\mu \mu _{3}+\nu \nu _{3}+\pi \pi _{3}=0$
${\text{that is }}\cos(\delta ,\delta _{1})={\frac {\varkappa }{\omega ^{2}q_{1}d}},\ \cos(\delta ,\delta _{2})=\cos(\delta ,\delta _{3})=0.$

5')

According to (4'), the three directions δ_{1}, δ_{2}, δ_{3} are perpendicular to each other, according to (5') $\delta _{1}$ falls into δ, then it must be:
$\mu =\mu _{1},\ \nu =\nu _{1},\ \pi =\pi _{1}\ {\text{ and }}\ {\frac {\varkappa }{\omega ^{2}q_{1}d}}=1.$

6)
