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

If we substitute in $U, V, W,$ respectively,

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

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

and this is fulfilled, because we have:

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

if there exist the following new equations:

 $\begin{array}{rl} 1-\omega^{2}(a^{2}+b^{2}+c^{2}) & =m_{1}^{2}+n_{1}^{2}+p_{1}^{2}-\frac{\alpha^{2}}{\omega^{2}}\\ & =m_{2}^{2}+n_{2}^{2}+p_{2}^{2}-\frac{\beta^{2}}{\omega^{2}}\\ & =m_{3}^{2}+n_{3}^{2}+p_{3}^{2}-\frac{\gamma^{2}}{\omega^{2}} \end{array}$ 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.