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

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

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).

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 it indeed has to be:

if there exist the following new equations:

