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

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):

${\begin{aligned}\xi _{1}&=x_{1}-\varkappa t\\\eta _{1}&=y_{1}q\\\zeta _{1}&=z_{1}q\\\tau &=t-{\frac {\varkappa x_{1}}{\omega ^{2}}}{,}{\text{ where }}q={\sqrt {1-{\frac {\varkappa ^{2}}{\omega ^{2}}}}}\end{aligned}}$[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.