# Page:Ueber das Doppler'sche Princip.djvu/3

 ${\displaystyle {\frac {\beta \gamma }{\omega ^{2}}}=m_{2}m_{3}+n_{2}n_{3}+p_{2}p_{3}}$ ${\displaystyle {\frac {\gamma \alpha }{\omega ^{2}}}=m_{3}m_{1}+n_{3}n_{1}+p_{3}p_{1}}$ ${\displaystyle {\frac {\alpha \beta }{\omega ^{2}}}=m_{1}m_{2}+n_{1}n_{2}+p_{1}p_{2}}$ 4)
 ${\displaystyle {\frac {\alpha }{\omega ^{2}}}=am_{1}+bn_{1}+cp_{1}}$ ${\displaystyle {\frac {\beta }{\omega ^{2}}}=am_{2}+bn_{2}+cp_{2}}$ ${\displaystyle {\frac {\gamma }{\omega ^{2}}}=am_{3}+bn_{3}+cp_{3}}$ 5)

If we take ${\displaystyle \alpha \beta \gamma }$ 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 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

${\displaystyle {\begin{array}{clcclcclcclc}m_{h}^{2}+n_{h}^{2}+p_{h}^{2}&=&q_{h}^{2},&m_{h}/q_{h}&=&\mu _{h},&n_{h}/q_{h}&=&\nu _{h},&p_{h}/q_{h}&=&\pi _{h}\\\\a^{2}+b^{2}+c^{2}&=&d^{2},&a/d&=&\mu ,&b/d&=&\nu ,&c/d&=&\pi ,\end{array}}}$

then μ, ν, π are the direction cosines of 4 directions, which we will denote by δ1, δ2, δ3 and δ, against the system X1, Y1, Z1.

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

 ${\displaystyle 1-\omega ^{2}d^{2}=q_{1}^{2}-{\frac {\varkappa ^{2}}{\omega ^{2}}}=q_{2}^{2}=q_{3}^{2}}$ 3')

${\displaystyle \mu _{2}\mu _{3}+\nu _{2}\nu _{3}+\pi _{2}\pi _{3}=\mu _{3}\mu _{1}+\nu _{3}\nu _{1}+\pi _{3}\pi _{1}=\mu _{1}\mu _{2}+\nu _{1}\nu _{2}+\pi _{1}\pi _{2}=0}$

 ${\displaystyle {\text{that is }}\cos(\delta _{2},\delta _{3})=\cos(\delta _{3},\delta _{1})=\cos(\delta _{1},\delta _{2})=0}$ 4')

${\displaystyle \mu \mu _{1}+\nu \nu _{1}+\pi \pi _{1}={\frac {\varkappa }{\omega ^{2}q_{1}d}},\ \mu \mu _{2}+\nu \nu _{2}+\pi \pi _{2}+\mu \mu _{3}+\nu \nu _{3}+\pi \pi _{3}=0}$

 ${\displaystyle {\text{that is }}\cos(\delta ,\delta _{1})={\frac {\varkappa }{\omega ^{2}q_{1}d}},\ \cos(\delta ,\delta _{2})=\cos(\delta ,\delta _{3})=0.}$ 5')

According to (4'), the three directions δ1, δ2, δ3 are perpendicular to each other, according to (5') ${\displaystyle \delta _{1}}$ falls into δ, then it must be:

 ${\displaystyle \mu =\mu _{1},\ \nu =\nu _{1},\ \pi =\pi _{1}\ {\text{ and }}\ {\frac {\varkappa }{\omega ^{2}q_{1}d}}=1.}$ 6)