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 δ_{1}, δ_{2}, δ_{3} fall into the coordinate axes X_{1}, Y_{1}, Z_{1}, that is

9) |

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

^{[AU 1]} |
10) |

The condition (1') is in this case

which can easily be exchanged with

10') |

This states, that in U the arguments *x* and *t* only may occur in connection with , 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 X_{1}, Y_{1}, Z_{1} to the general *X, Y, Z*, which is connected with the preceding by the relations

11) |

- ↑ This is, except the factor
*q*which is irrelevant for the application, exactly the Lorentz transformation of the year 1904.