Page:PoincareDynamiqueJuin.djvu/3

These formulas differ somewhat from those which had been found by Lorentz.

Let now $X,Y,Z$ and $X',Y',Z'$ the three components of force before and after transformation, and the force is expressed in unit volume; I found

(3)	$X^{\prime }={\frac {k}{l^{3}}}(X+\epsilon \Sigma X\xi ),\quad Y^{\prime }={\frac {Y}{l^{3}}},\quad Z^{\prime }={\frac {Z}{l^{3}}}$

These formulas are slightly different from those of Lorentz; the additional term in $\Sigma X\xi$ reminds us on a result previously obtained by Liénard.

If we now denote by $X_{1},\ Y_{1},\ Z_{1}$ and $X'_{1},\ Y'_{1},\ Z'_{1}$ the components of a force, not referred to unit volume, but to unit mass of the electron, we obtain

(4)	$X_{1}^{\prime }={\frac {k}{l^{3}}}{\frac {\rho }{\rho ^{\prime }}}(X_{1}+\epsilon \Sigma X_{1}\xi ),\quad Y_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Y_{1}}{l^{3}}},\quad Z_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}{\frac {Z_{1}}{l^{3}}}$

Lorentz was also led to assume that the electron in motion takes the form of an oblate spheroid; this is also the hypothesis made by Langevin, however, while Lorentz assumed that two axes of the ellipsoid remain constant, which is consistent with the hypothesis $l=1$, Langevin assumed that the volume remains constant. Both authors have shown that these two hypotheses are consistent with the experiments of Kaufmann, as well as the original hypothesis of Abraham (spherical electron). The hypothesis of Langevin would have the advantage that it is self-sufficient, because it suffices to regard the electron as deformable and incompressible, and to explain that it takes an ellipsoidal shape when it moves. But I show, in agreement with Lorentz, that it is incapable to accord with the impossibility of an experiment showing the absolute motion. As I have said, this is because $l=1$ is the only case for which all the Lorentz transformations form a group.

But with the hypothesis of Lorentz, the agreement between the formulas does not occur all alone; we obtain at the same time a possible explanation for the contraction of the electron, assuming that the electron, deformable and compressible, is subjected to a constant external pressure whose work is proportional to volume changes.

I show, by applying the principle of least action,