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practically sufficient to determine the relation between the two systems of measurement in certain cases, as, for instance, when the B's are moving with a uniform velocity relative to the A's.^[1]

It is found that in the case of uniform relative motion the units of length and time in the two systems are different, and that two events occurring at different points of space may appear to be simultaneous according to measurements made by the A's, and not appear to be simultaneous according to measurements made by the B's. Also, the shape of a body is theoretically different according to the two series of measurements, but the difference is so very slight as to be unnoticeable.

The fact that the electromagnetic equations have the same form in the two systems of coordinates, indicates, that as far as our observations of electromagnetic phenomena are concerned, a uniform motion of a system of observers would remain undetected. This, of course, is in accordance with the view that position and motion are purely relative, and that the term absolute motion is meaningless.

1. The transformation for the case of uniform velocity was given by Voigt, Larmor and Lorentz, it is

   $x'={\frac {x-vt}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ y'=y,\ z'=z,\ t'={\frac {t-{\frac {vx}{c^{2}}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}.$

   This transformation makes

   $\left(x'_{1}-x'_{2}\right)^{2}+\left(y'_{1}-y'_{2}\right)^{2}+\left(z'_{1}-z'_{2}\right)^{2}-c^{2}\left(t'_{1}-t'_{2}\right)^{2}=0,$

   a consequence of

   $\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}+\left(z_{1}-z_{2}\right)^{2}-c^{2}\left(t_{1}-t_{2}\right)^{2}=0.$

   It can be shown that any transformation of coordinates which leaves the last equation unaltered is such that the electrodynamical equations are unaltered in form. See papers by E. Cunningham and the author in the Proceedings of the London Mathematical Society (1910).