Page:Ueber das Doppler'sche Princip.djvu/2

If we substitute in ${\displaystyle U,V,W,}$ respectively,

 ${\displaystyle {\begin{array}{l}x{\text{ by }}\xi =xm_{1}+yn_{1}+zp_{1}-\alpha t\\y{\text{ by }}\eta =xm_{2}+yn_{2}+zp_{2}-\beta t\\z{\text{ by }}\xi =xm_{3}+yn_{3}+zp_{3}-\gamma t\\t{\text{ by }}\tau =t-(ax+by+cz)\\\end{array}}}$ 2)

and describe the resulting functions, respectively, with (U), (V), (W), then by u = (U), v = (V), w = (W) it is possible to comply with (1). [AU 1]

For example, we obtain for the first of them:

${\displaystyle {\frac {\partial ^{2}(U)}{\partial \tau ^{2}}}\left(1-\omega ^{2}\left(a^{2}+b^{2}+c^{2}\right)\right)=\omega ^{2}\left\{{\frac {\partial ^{2}(U)}{\partial \xi ^{2}}}\left(m_{1}^{2}+n_{1}^{2}+p_{1}^{2}-{\frac {\alpha ^{2}}{\omega ^{2}}}\right)\right.}$
${\displaystyle +{\frac {\partial ^{2}(U)}{\partial \eta ^{2}}}\left(m_{2}^{2}+n_{2}^{2}+p_{2}^{2}-{\frac {\beta ^{2}}{\omega ^{2}}}\right)+{\frac {\partial ^{2}(U)}{\partial \zeta ^{2}}}\left(m_{3}^{2}+n_{3}^{2}+3_{3}^{2}-{\frac {\gamma ^{2}}{\omega ^{2}}}\right)}$
${\displaystyle +2{\frac {\partial ^{2}(U)}{\partial \eta \ \partial \zeta }}\left(m_{2}m_{3}+n_{2}n_{3}+p_{2}p_{3}-{\frac {\beta \gamma }{\omega ^{2}}}\right)}$
${\displaystyle +2{\frac {\partial ^{2}(U)}{\partial \zeta \ \partial \xi }}\left(m_{3}m_{1}+n_{3}n_{1}+p_{3}p_{1}-{\frac {\gamma \alpha }{\omega ^{2}}}\right)}$
${\displaystyle +2{\frac {\partial ^{2}(U)}{\partial \xi \ \partial \eta }}\left(m_{1}m_{2}+n_{1}n_{2}+p_{1}p_{2}-{\frac {\alpha \beta }{\omega ^{2}}}\right)}$
${\displaystyle -2{\frac {\partial ^{2}(U)}{\partial \tau \ \partial \xi }}\left(am_{1}+bn_{1}+cp_{1}-{\frac {\alpha }{\omega ^{2}}}\right)}$
${\displaystyle -2{\frac {\partial ^{2}(U)}{\partial \tau \ \partial \eta }}\left(am_{2}+bn_{2}+cp_{2}-{\frac {\beta }{\omega ^{2}}}\right)}$
${\displaystyle \left.-2{\frac {\partial ^{2}(U)}{\partial \tau \ \partial \zeta }}\left(am_{3}+bn_{3}+cp_{3}-{\frac {\gamma }{\omega ^{2}}}\right)\right\}}$

and this is fulfilled, because it indeed has to be:

${\displaystyle {\frac {\partial ^{2}(U)}{\partial \tau ^{2}}}=\omega ^{2}\left({\frac {\partial ^{2}(U)}{\partial \xi ^{2}}}+{\frac {\partial ^{2}(U)}{\partial \eta ^{2}}}+{\frac {\partial ^{2}(U)}{\partial \zeta ^{2}}}\right),}$

if there exist the following new equations:

 ${\displaystyle {\begin{array}{rl}1-\omega ^{2}(a^{2}+b^{2}+c^{2})&=m_{1}^{2}+n_{1}^{2}+p_{1}^{2}-{\frac {\alpha ^{2}}{\omega ^{2}}}\\&=m_{2}^{2}+n_{2}^{2}+p_{2}^{2}-{\frac {\beta ^{2}}{\omega ^{2}}}\\&=m_{3}^{2}+n_{3}^{2}+p_{3}^{2}-{\frac {\gamma ^{2}}{\omega ^{2}}}\end{array}}}$ 3)
1. Due to the same order of all parts of equations (1), the right-hand sides of the substitution formulas (2) can be multiplied by a common factor, without changing the results.