Page:PoincareDynamiqueJuillet.djvu/32

This is why Abraham gave ${\tfrac {dD}{dV}}$ the name longitudinal mass and ${\tfrac {D}{V}}$ the name transverse mass; recall that $D={\tfrac {dH}{dV}}.$

In the hypothesis of Lorentz, we have:

$D=-{\frac {dH}{dV}}=-{\frac {\partial H}{\partial V}},$

${\tfrac {\partial H}{\partial V}}$ represent the derivative with respect to V, after r and θ were replaced by their values as functions of V from the first two equations (1); we will also have, after the substitution,

$H=+A{\sqrt {1-V^{2}}}.$

We choose units so that the constant factor A is equal to 1, and I pose ${\sqrt {1-V^{2}}}=h$, hence:

$H=+h,\quad D={\frac {V}{h}},\quad {\frac {dD}{dV}}=h^{-3},\quad {\frac {dD}{dV}}{\frac {1}{V^{2}}}-{\frac {D}{V^{3}}}=h^{-3}.$

We will pose again:

$M=V{\frac {dV}{dt}}=\sum \xi {\frac {d\xi }{dt}},\quad X_{1}=\int Xd\tau$

and we find the equation for quasi-stationary motion:

(5)	$h^{-1}{\frac {d\xi }{dt}}+h^{-3}\xi M=X_{1}.$

Let's see what happens to these equations by the Lorentz transformation. We will pose: $1+\xi \epsilon =\mu$, and we have first:

$\mu \xi ^{\prime }=\xi +\epsilon ,\quad \mu \eta ^{\prime }={\frac {\eta }{k}},\quad \mu \zeta ^{\prime }={\frac {\zeta }{k}}$

from which we derive easily

$\mu h^{\prime }={\frac {h}{k}}.$

We also have

$dt'=k\ \mu \ dt$

where:

${\frac {d\xi ^{\prime }}{dt^{\prime }}}={\frac {d\xi }{dt}}{\frac {1}{k^{3}\mu ^{3}}},\quad {\frac {d\eta ^{\prime }}{dt^{\prime }}}={\frac {d\eta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\eta \epsilon }{k^{2}\mu ^{3}}},\quad {\frac {d\zeta ^{\prime }}{dt^{\prime }}}={\frac {d\zeta }{dt}}{\frac {1}{k^{2}\mu ^{2}}}-{\frac {d\xi }{dt}}{\frac {\zeta \epsilon }{k^{2}\mu ^{3}}}$

where again:

$M^{\prime }={\frac {d\xi }{dt}}{\frac {\epsilon h^{2}}{k^{3}\mu ^{4}}}+{\frac {M}{k^{3}\mu ^{3}}}$

and

(6)	$h^{\prime -1}{\frac {d\xi ^{\prime }}{dt^{\prime }}}+h^{\prime -3}\xi ^{\prime }M^{\prime }=\left[h^{-1}{\frac {d\xi }{dt}}+h^{-3}(\xi +\epsilon )M\right]\mu ^{-1},$