Page:PoincareDynamiqueJuillet.djvu/26

This is in contradiction with the results of § 4 and with the result obtained by Lorentz by another way. That is the contradiction which is to be explained.

Before addressing this explanation, I note that whatever is the hypotheses we have adopted

$H=A+B-C={\frac {l}{k}}(A^{\prime }+B^{\prime }),$

or, because of C' = 0,

(3)	$H={\frac {l}{k}}H^{\prime },$

We can compare the result of the equation J = J' obtained in § 3.

We have indeed:

$J=\int H\ dt,\quad J^{\prime }=\int H^{\prime }\ dt^{\prime }.$

We observe that the state of the system depends only on x + εt, y and z, that is to say on x', y', z', and we have:

$t^{\prime }={\frac {l}{k}}t+\epsilon x^{\prime }$

(4)	$dt^{\prime }={\frac {l}{k}}dt.$

By comparing equations (3) and (4) we find J = J'.

Let us consider any hypothesis, which may be either that of Lorentz, or that of Abraham, or that of Langevin, or an intermediate hypothesis.

Let

$r,\ \theta r,\ \theta r$

the three axes of the real electron; that of the ideal electron will be:

$klr,\ \theta lr,\ \theta lr$

Then A' + B' is the electrostatic energy of an ellipsoid with axes klr, θlr, θlr.

Let us suppose that the electricity is spread on the surface of the electron as it is known from an inductor, or uniformly distributed within the electron; than this energy will be of the form:

$A^{\prime }+B^{\prime }={\frac {\varphi \left({\frac {\theta }{k}}\right)}{klr}},$

where φ is a known function.

The hypothesis of Abraham is to assume:

$r=const.,\ \theta =1.$

That of Lorentz:

$l=1,\ kr=const.,\ \theta =k.$