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that the transformations, with respect to which the laws of nature shall be invariant, are "Lorentz-transformations."^[1] A Lorentz-transformation is defined by a modulus $q$ and an axis. Taking the latter as the axis of $x$, the transformation-formulæ are—

$x'={\frac {x-qct}{\sqrt {1-q^{2}}}},\quad y'=y,\quad z'=z,\quad ct'={\frac {ct-qx}{\sqrt {1-q^{2}}}},$

where $c$ is a universal constant, which, according to the electromagnetic theory, is equal to the velocity of light in free space. In Newtonian mechanics the value of this constant is $c=\infty$. Putting, then, $q=v/c$, so that $q=0$, the Lorentz-transformation degenerates into a "Newton-transformation,"—

$x'=x-vt,\quad y'=y,\quad z'=z,\quad t'=t$

To both Lorentz- and Newton-transformations may be added an arbitrary orthogonal transformation of coordinates.

The physical meaning of the principle has been very clearly explained in these pages by Messrs. Plummer and Whittaker,^[2] and need not be repeated here. The mathematical formulæ are all that is required for our purpose.

The literature of the subject is very extensive, and it is hardly possible for an outsider to be even superficially acquainted with it. Also I do not claim originality for any of the formulæ or results given below. The starting-point of my investigations has been the papers by Poincaré and Minkowski.^[3] The manner in which the equations of motion are derived below is entirely derived from the last section of Poincaré’s paper. I also owe much to conversations with and advice from my colleague Professor Lorentz.

2. Let there be two systems of reference:—

⁠ the "general" system $(x',y',z',t')$,
and the "special" system $(x,y,z,t)$.

The first is an absolutely arbitrary system of reference. The

1. This name has been first introduced by Poincaré in the paper quoted below.
2. H. C. Plummer. "On the Theory of Aberration and the Principle of Relativity" (M.N., Jan, 1910). E.T.W.: "Recent Researches on Space, Time, and Force" (Report of the Council, M.N., Feb. 1910). Both authors make free use of the word "aether." As there are many physicists nowadays who are inclined to abandon the aether altogether, it may be well to point out that the principle of relativity is essentially independent of the concept of an aether, and, indeed, is considered by some to lead to a negation of its existence. Astronomers have nothing to do with the aether, and it need not concern them whether it exists or not. All Mr. Plummer’s results remain true, and retain their full value, if the "aether" is eliminated from his terminology. And also in Mr. Whittaker’s note the word "aether" is not essential, except, of course, from an historical point of view.
3. Poincaré: "Sur la dynamique de l'électron," Rendiconti del circolo matematico di Palermo, vol. xxi. p. 129 (Dec. 1905). Minkowski: "Die Grundgleichungen für die electromagnetische Vorgänge in bewegten Körpern," Göttinger Nachrichten, Math. physik. Klasse, 1908, page 53.