Page:PoincareDynamiqueJuillet.djvu/6

We thus find the formulas:

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

but the value of ρ' differs.

It is important to note that formulas (4) and (4^bis) satisfy the continuity condition

${\frac {d\rho ^{\prime }}{dt^{\prime }}}+\sum {\frac {d\rho ^{\prime }\xi ^{\prime }}{dx^{\prime }}}=0.$

Indeed, let λ be an undetermined quantity and D the functional determinant

(5)	$t+\lambda \rho ,\ x+\lambda \rho \xi ,\ x+\lambda \rho \eta ,\ z+\lambda \rho \zeta$

with respect to t, x, y, z. We will have:

$D=D_{0}+D_{1}\lambda +D_{2}\lambda ^{2}+D_{3}\lambda ^{3}+D_{4}\lambda ^{4}\,$

with $D_{0}=1,\,D_{1}={\frac {d\rho }{dt}}+\sum {\frac {d\rho \xi }{dx}}=0.$

Let $\lambda '=l^{2}\lambda$, we see that the four functions

5bis	$t^{\prime }+\lambda ^{\prime }\rho ^{\prime },\ x^{\prime }+\lambda ^{\prime }\rho ^{\prime }\xi ^{\prime },\ y^{\prime }+\lambda ^{\prime }\rho ^{\prime }\eta ^{\prime },\ z^{\prime }+\lambda ^{\prime }\rho ^{\prime }\zeta ^{\prime }$

are related to the functions (5) by the same linear relations as the old variables to the new variables. Then, if we denote by D' the functional determinant of the functions (5^bis) in relation to the new variables, we have:

$D^{\prime }=D,\ D^{\prime }=D_{0}^{\prime }+D_{1}^{\prime }\lambda ^{\prime }+\ldots +D_{4}^{\prime }\lambda ^{\prime 4},$

where:

$D_{0}^{\prime }=D_{0}=1,\ D_{1}^{\prime }=l^{-2}D_{1}=0={\frac {d\rho ^{\prime }}{dt^{\prime }}}+\sum {\frac {d\rho ^{\prime }\xi ^{\prime }}{dx^{\prime }}}.$ C. Q. F. D

With the hypothesis of Lorentz, this condition is not satisfied, since ρ' has not the same value.

We will define the new potentials, vector and scalar, in order to satisfy the conditions

(6)	$\square ^{\prime }\psi ^{\prime }=-\rho ^{\prime },\quad \square ^{\prime }F^{\prime }=-\rho ^{\prime }\xi ^{\prime }.$

Then we obtain from this:

(7)	$\psi ^{\prime }={\frac {k}{l}}(\psi +\epsilon F),\ F^{\prime }={\frac {k}{l}}(F+\epsilon \psi ),\ G^{\prime }={\frac {1}{l}}G,\ H^{\prime }={\frac {1}{l}}H.$

These formulas differ significantly from those of Lorentz, but the difference is ultimately due to the definitions.

We will choose the new electric and magnetic fields so as to satisfy the equations:

(8)	$f^{\prime }=-{\frac {dF^{\prime }}{dt^{\prime }}}-{\frac {d\psi ^{\prime }}{dx^{\prime }}},\quad \alpha ^{\prime }={\frac {dH^{\prime }}{dy^{\prime }}}-{\frac {dG^{\prime }}{dz^{\prime }}}.$