Page:PoincareDynamiqueJuillet.djvu/19

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

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:

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 2d and 3rd 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:

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