Page:CunninghamExtension.djvu/4

80 with the space (X, Y, Z, icT) of the other: inasmuch as

must be a consequence of

We may, however, prove that no conformal transformation in this space exists which will transform every point at rest in the space (x, y, z) into a point moving with angular velocity about a fixed line in the (XYZ) space. Thus, as in material dynamics, the equations of electrodynamics will not be preserved in the same form for a set of axes rotating uniformly relatively to the space in which they are satisfied by actual phenomena.

2. The following relations arising immediately from the Lorentz-Einstein transformation are used in what follows.

The velocities of a moving point in the two systems are connected thus:

{{MathForm2|(2)|$$\left.\begin{array}{l} W_{X}=\left(w_{x}-v\right)/\left(1-\frac{vw_{x}}{c^{2}}\right)\\ \\W_{N}=w_{n}/\left(1-\frac{vw_{x}}{c^{2}}\right)\end{array}\right\} ,$$}}

the suffixes n and N denoting components in any direction perpendicular to v.

The relative coordinates of two moving points (x, y, z), (x', y', z') :

{{MathForm2|(3)|$$\left.\begin{array}{l} X'-X=(x'-x)/\beta\left(1-\frac{vw'_{x}}{c^{2}}\right)\\ \\N'-N=(n'-n)+vw'_{n}(x'-x)/c^{2}\left(1-\frac{vw'_{x}}{c^{2}}\right)\end{array}\right\} ,$$}}

The relative velocity of two moving points:

{{MathForm2|(4)|$$\left.\begin{array}{l} W'_{X}-W_{X}=(w'_{x}-w_{x})/\beta^{2}\left(1-\frac{vw_{x}}{c^{2}}\right)\left(1-\frac{vw'_{x}}{c^{2}}\right)\\ \\W'_{N}-W_{N}=(w'_{n}-w_{n})/\beta\left(1-\frac{vw_{x}}{c^{2}}\right)-vw'_{n}\left(w'_{x}-w_{x}\right)/\beta c^{2}\left(1-\frac{vw_{x}}{c^{2}}\right)\left(1-\frac{vw'_{x}}{c^{2}}\right)\end{array}\right\} .$$}}