Page:CunninghamExtension.djvu/4

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

$\delta X^{2}+\delta Y^{2}+\delta Z^{2}-c^{2}\delta T^{2}=0$

must be a consequence of

$\delta x^{2}+\delta y^{2}+\delta z^{2}-c^{2}\delta t^{2}=0.$

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.

I.

Some considerations concerning the Lorentz-Einstein Transformation.

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:

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

(2)

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')^[1]:

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

(3)

The relative velocity of two moving points:

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

(4)

---

1. Inasmuch as the relative coordinates are changing, it is necessary to specify exactly when they are measured. The expressions given are the values at the instant at which the second point is at (x'y'z'). n,n',..., here stand for either coordinate perpendicular to x, x', ....