Page:Ueber das Doppler'sche Princip.djvu/4

 Substituting this in (3'), q1, q2, q3 are determined.

At first we obtain, since only positive signs are meaningful:

$$q_{1}=1\text{ or }\frac{\varkappa}{\omega}$$

$$d=\frac{\varkappa}{\omega^{2}}\text{ or }\frac{1}{\omega}.$$

I will only use the first solution, the second is of no interest; it follows from it:

Consequently, we can write equations (2):

where for &mu;h, νh, πh no more other conditions apply than those which result from their meaning as direction cosines of three successive perpendicular but otherwise quite arbitrary directions.

Therefore, the aggregates designated by $$a_{1}\ b_{1}\ c_{1}$$ can be considered as the coordinates of the point $$x_{1}\ y_{1}\ z_{1}$$ in relation to a coordinate system, which falls into the direction $$\delta_{1}\ \delta_{2}\ \delta_{3}$$.

Any such system &mu;h, νh, πh gives a solution (U), (V), (W) from given U, V, W. If U, V, W adopt on a surface f(x, y, z) = 0 the given values $$\overline{U}$$, $$\overline{V}$$, $$\overline{W}$$, so (U), (V), (W) from those derivable $$(\overline{U}), (\overline{V}), (\overline{W})$$ to the surface $$(f)=f(\overline{\xi_{1}},\overline{\eta_{1}},\overline{\zeta_{1}})=0$$, which because of the values of ξ1, η1, ζ1 has the property to move with uniform velocity ϰ parallel to a direction δ1 or A given by direction cosines ϰ. The solutions (U), (V), (W) give thus the laws by which certain surfaces in progressive motion are shining, if they only comply with the condition

$$\frac{(\partial U)}{\partial x}+\frac{\partial (V)}{\partial y}+\frac{\partial (W)}{\partial z}=0$$