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Mar. 1911. that the transformations, with respect to which the laws of nature shall be invariant, are "Lorentz-transformations." A Lorentz-transformation is defined by a modulus $$q$$ and an axis. Taking the latter as the axis of $$x$$, the transformation-formulæ are—

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,"—

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, 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. 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