Page:PoincareDynamiqueJuillet.djvu/19

We know we can integrate by the retarded potentials and we have:

(2)	$\psi ={\frac {1}{4\pi }}\int {\frac {\rho _{1}d\tau }{r}},\quad F={\frac {1}{4\pi }}\int {\frac {\rho _{1}\xi _{1}d\tau _{1}}{r}}.$

In these formulas we have:

$d\tau _{1}=dx_{1}dy_{1}dz_{1},\quad r^{2}=(x-x_{1})^{2}+(y-y_{1})^{2}+(z-z_{1})^{2},$

whereas ρ₁ and ξ₁ are the values of ρ and ξ at the point x₁, y₁, z₁ and the instant

$t_{1}=t-r\,$

x₀, y₀, z₀ being coordinates of a molecule of the electron at the instant t;

$x_{1}=x_{0}+U,\ y_{1}=y_{0}+V,\ z_{1}=z_{0}+W$

being its coordinates at the instant t₁;

U, V, W are functions of x₀, y₀, z₀, so that we can write:

$dx_{1}=dx_{0}+{\frac {dU}{dx_{0}}}dx_{0}+{\frac {dU}{dy_{0}}}dy_{0}+{\frac {dU}{dz_{0}}}dz_{0}+\xi _{1}dt_{1};$

and if we assume t to be constant, as well as x, y and z:

$dt_{1}=+\sum {\frac {x-x_{1}}{r}}dx_{1}.$

We can therefore write:

$dx_{1}\left(1+\xi _{1}{\frac {x_{1}-x}{r}}\right)+dy_{1}\xi _{1}{\frac {y_{1}-y}{r}}+dz_{1}\xi _{1}{\frac {z_{1}-z}{r}}=dx_{0}\left(1+{\frac {dU}{dx_{0}}}\right)+dy_{0}{\frac {dU}{dy_{0}}}+dz_{0}{\frac {dU}{dz_{0}}}$

so that the other two equations can deduced by circular permutation.

We therefore have:

(3)	$d\tau _{1}\mid 1+\xi _{1}{\frac {x_{1}-x}{r}},\quad \xi _{1}{\frac {y_{1}-y}{r}},\quad \xi _{1}{\frac {z_{1}-z}{r}}\mid =d\tau _{0}\mid 1+{\frac {dU}{dx_{0}}},\quad {\frac {dU}{dy_{0}}},\quad {\frac {dU}{dz_{0}}}\mid ,$

we set

$d\tau _{0}=dx_{0}dy_{0}dz_{0}\,$

Consider the determinants that appear in both sides of (3) and at the begin of the first part; if we seek to develop, we see that the terms of the 2^d and 3^rd degree from ξ₁, η₁, ζ₁ disappear and that the determinant is equal to

$1+\xi _{1}{\frac {x_{1}-x}{r}}+\eta _{1}{\frac {y_{1}-y}{r}}+\zeta _{1}{\frac {z_{1}-z}{r}}=1+\omega ,$

ω designates the radial component of the velocity ξ₁, η₁, ζ₁, that is to say, the component directed along the radius vector indicating from point x, y, t to point x₁, y₁, z₁.

In order to obtain the second determinant, I look at the coordinates of different molecules of the electron at instant t', which is the same for all molecules, but in such a way that for the molecule considered we have $t_{1}=t'_{1}$. The coordinates of a molecule will then be:

$x'_{1}=x_{0}+U',\ y'_{1}=y_{0}+V',\ z'_{1}=z_{0}+W'$