Page:Ueber das Doppler'sche Princip.djvu/5

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The two surfaces f = 0 and (f) = 0 have identical forms only if q = 1, i.e. ϰ is so small against ω, that ϰ² can be neglected with respect to ω². If this is the case, then they differ only by their position against the coordinate axes. By appropriate use of the arbitrary constants and the functions U, V, W we can obtain vivid special cases. By coordinate transformation we are lead to a (at least formally) general case, in which the shift of the surface is not parallel to the A-axis parallel, but directed in an arbitrary way.

We follow the special case, in which the three directions δ1, δ2, δ3 fall into the coordinate axes X1, Y1, Z1, that is

\mu_{1}=\nu_{2}=\pi_{3}=1{,}

\mu_{2}=\mu_{3}=\nu_{1}=\nu_{2}=\pi_{1}=\pi_{3}=0

9)

Then it is given, in a very simple and natural way, and formally identical with (8):

\begin{align}
\xi_{1} & =x_{1}-\varkappa t\\
\eta_{1} & =y_{1}q\\
\zeta_{1} & =z_{1}q\\
\tau & =t-\frac{\varkappa x_{1}}{\omega^{2}}{,}\text{ where }q=\sqrt{1-\frac{\varkappa^{2}}{\omega^{2}}}
\end{align}[AU 1] 10)

The condition (1') is in this case

(1-q)\frac{\partial(U)}{\partial\xi}=\frac{\varkappa}{\omega^{2}}\frac{\partial(U)}{\partial\tau}

which can easily be exchanged with

(1-q)\frac{\partial U}{\partial x}=\frac{\varkappa}{\omega^{2}}\frac{\partial U}{\partial t}. 10')

This states, that in U the arguments x and t only may occur in connection with (1-q)t+\frac{\varkappa x}{\omega^{2}}, or not at all. The latter is the case if U = 0, that is, when the propagated vibrations are everywhere normal to the direction of translation of the illuminating surface.

If we pass from the assumed special co-ordinate system X1, Y1, Z1 to the general X, Y, Z, which is connected with the preceding by the relations

x_{1}=x\alpha_{1}+y\beta_{1}+z\gamma_{1}

y_{1}=x\alpha_{2}+y\beta_{2}+z\gamma_{2}

z_{1}=x\alpha_{3}+y\beta_{3}+z\gamma_{3},

11)
  1. This is, except the factor q which is irrelevant for the application, exactly the Lorentz transformation of the year 1904.