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which might demonstrate the absolute motion of the Earth. But if the impossibility of such a finding is considered highly probable, one can predict that these experiments, if they can ever be conducted, will give a negative result. Lorentz has sought to supplement and amend his hypothesis so as to bring it into accord with the postulate of the complete impossibility of determining absolute motion. This he managed to do in his article entitled Electromagnetic phenomena in a system moving with any velocity smaller than that of Light (Proceedings de l’Académie d’Amsterdam, May 27, 1904).

The importance of this question made me determined to return to it; and the results I obtained are in agreement on all important points with those of Lorentz; I was only led to modify and complete them in a few points of detail.

The essential point, established by Lorentz, is that the electromagnetic field equations are not altered by a certain transformation (which I shall call after the name of Lorentz), which has the following form:

(1)	$x^{\prime }=kl(x+\epsilon t),\quad y^{\prime }=ly,\quad z^{\prime }=lz,\quad t^{\prime }=kl(t+\epsilon x)$

$x,y,z$ are the coordinates and $t$ the time before the transformation, $x',y',z'$ and $t'$ after the transformation. Moreover, $\epsilon$ is a constant which defines the transformation

$k={\frac {1}{\sqrt {1-\epsilon ^{2}}}}$,

and $l$ is an arbitrary function of $\epsilon$. One can see that in this transformation the $x$-axis plays a particular role, but one can obviously construct a transformation in which this role would be played by any straight line through the origin. The sum of all these transformations, together with the set of all rotations of space, must form a group; but for this to occur, we need $l=1$; so one is forced to suppose $l=1$ and this is a consequence that Lorentz has obtained by another way.

Let $\rho$ the electric density of the electron, $\xi ,\eta ,\zeta$ the velocity before the transformation; we obtain for the same quantities $\rho ',\xi ',\eta ',\zeta '$ after processing

(2)	$\rho ^{\prime }={\frac {k}{l^{3}}}\rho (1+\epsilon \xi ),\quad \rho ^{\prime }\xi ^{\prime }={\frac {k}{l^{3}}}\rho (\xi +\epsilon ),\quad \rho ^{\prime }\eta ^{\prime }={\frac {\rho \eta }{l^{3}}},\ \quad \rho ^{\prime }\zeta ^{\prime }={\frac {\rho \zeta }{l^{3}}}$