Page:PoincareDynamiqueJuillet.djvu/31

by light to travel from one point to another of the electron; in other words, H depend not only on ξ, η, ζ, r, θ, but on their derivatives of all orders with respect to time.

Well, the motion is said to be quasi-stationary when the partial derivatives of H with respect to the successive derivatives of ξ, η, ζ, r, θ are negligible compared to the partial derivatives of H with respect to the quantities ξ, η, ζ, r, θ themselves.

The equations of such a motion can be written:

(1)	${\begin{cases}&{\frac {dH}{d\theta }}+{\frac {dF}{d\theta }}={\frac {dH}{dr}}+{\frac {dF}{dr}}=0,\\\\&{\frac {d}{dt}}{\frac {dH}{d\xi }}=-\int Xd\tau ,\quad {\frac {d}{dt}}{\frac {dH}{d\eta }}=-\int Yd\tau ,\quad {\frac {d}{dt}}{\frac {dH}{d\zeta }}=-\int Zd\tau .\end{cases}}$

In these equations, F has the same meaning as in the preceding §, X, Y, Z are the components of the force acting on the electron: this force is solely due to electric and magnetic fields produced by other electrons.

Note that H is independent of ξ η ζ through the combination

$V={\sqrt {\xi ^{2}+\eta ^{2}+\zeta ^{2}}},\,$

that is to say, the magnitude of the velocity; therefore we still call D the momentum:

${\frac {dH}{d\xi }}={\frac {dH}{dV}}{\frac {\xi }{V}}=-D{\frac {\xi }{V}},$

where:

(2)	$-{\frac {d}{dt}}{\frac {dH}{d\xi }}={\frac {D}{V}}{\frac {d\xi }{dt}}-D{\frac {\xi }{V^{2}}}{\frac {dV}{dt}}+{\frac {dD}{dV}}{\frac {\xi }{V}}{\frac {dV}{dt}},$

(2bis)

$-{\frac {d}{dt}}{\frac {dH}{d\eta }}={\frac {D}{V}}{\frac {d\eta }{dt}}-D{\frac {\eta }{V^{2}}}{\frac {dV}{dt}}+{\frac {dD}{dV}}{\frac {\eta }{V}}{\frac {dV}{dt}},$

with

(3)	$V{\frac {DV}{dt}}=\sum \xi {\frac {d\xi }{dt}}.$

If we take the current direction of the velocity as the x-axis, we get:

$\xi =V,\quad \eta =\zeta =0,\quad {\frac {d\xi }{dt}}={\frac {dV}{dt}};$

equations (2) and (2_bis) become:

$-{\frac {d}{dt}}{\frac {dH}{d\xi }}-{\frac {dD}{dV}}{\frac {d\xi }{dt}},\quad -{\frac {d}{dt}}{\frac {dH}{d\eta }}={\frac {D}{V}}{\frac {d\eta }{dt}}$

and the last three equations (1):

(4)	${\frac {dD}{dV}}{\frac {d\xi }{dt}}=\int Xd\tau ,\quad {\frac {D}{V}}{\frac {d\eta }{dt}}=\int Yd\tau ,\quad {\frac {D}{V}}{\frac {d\zeta }{dt}}=\int Zd\tau .$