Page:PoincareDynamiqueJuillet.djvu/35

or by posing

${\frac {\varphi (\xi )}{f(\xi )}}=\Omega (\xi )={\frac {D}{V{\frac {dD}{dV}}}},$

that is:

$\Omega \left({\frac {\xi +\epsilon }{1+\xi \epsilon }}\right)=\Omega (\xi ){\frac {1+\epsilon ^{2}}{(1+\xi \epsilon )^{2}}},$

an equation that must be satisfied for all values of ξ and ε. For ζ = 0 we find:

$\Omega (\epsilon )=\Omega (0)\left(1-\epsilon ^{2}\right),$

where:

$D=A\left({\frac {V}{\sqrt {1-V^{2}}}}\right)^{m},$

A is a constant, and I set $\Omega (0)={\frac {1}{m}}$.

We then find:

$\varphi (\xi )={\frac {A}{\xi }}\left({\frac {\xi }{\sqrt {1-\xi ^{2}}}}\right)^{m},\quad \varphi (\xi ^{\prime })={\frac {A\mu }{\xi +\epsilon }}\left({\frac {\xi +\epsilon }{{\sqrt {1-\xi ^{2}}}{\sqrt {1-\epsilon ^{2}}}}}\right)^{m}.$

Now $\varphi (\xi ^{\prime })=\varphi (\xi ){\frac {k\mu }{l^{2}}}$; so we have:

$(\xi +\epsilon )^{m-1}\left(1-\epsilon ^{2}\right)^{-{\frac {m}{2}}}=-\xi ^{m-1}\left(1-\epsilon ^{2}\right)^{-{\frac {1}{2}}}l^{-2}.$

As l should depend only on ε (since, if there are more electrons, l must be the same for all electrons whose velocities ξ may be different), this identity can take place only if we have:

$m=1,\ l=1.$

Thus Lorentz's hypothesis is the only one consistent with the inability to demonstrate absolute motion; if we accept this impossibility, we must admit that the moving electrons contract and become ellipsoids of revolution where two axes remain constant; it must be admitted, as we have shown in the previous §, the existence of an additional potential which is proportional to the volume of the electron.

The analysis of Lorentz is therefore fully confirmed, but we can better give us an account of the true reason of the fact which occupies us; and this reason must be sought in the considerations of § 4. The transformations that do not alter the equations of motion must form a group, and this can take place only if l = 1. As we do not recognize if an electron is at rest or in absolute motion, it is necessary that, when in motion, it undergoes a distortion that must be precisely that which imposes the corresponding transformation of the group.