Page:PoincareDynamiqueJuillet.djvu/43

 deduced from the law of, and when they are in relative rest, the value deduced from the equations (4).

Under the hypothesis of absolute rest, the first two invariants (7) must be reduced to

$$\sum X_{1}^{2},\quad\sum X_{1}x,$$

or by 's law at

$$\frac{1}{r^{4}},\quad-\frac{1}{r}$$

secondly, in hypothesis (A), the 2nd and 3rd of the invariants (5) become:

$$\frac{-r-\sum x\xi}{\sqrt{1-\sum\xi^{2}}},\quad\frac{-r-\sum x\xi_{1}}{\sqrt{1-\sum\xi_{1}^{2}}}$$

that is to say, for absolute rest, to

$$-r,\ -r$$.

We may therefore assume for example that the first two invariants (4) are reduced to

$$\frac{\left(1-\sum x\xi_{1}^{2}\right)^{2}}{\left(r+\sum x\xi_{1}\right)^{4}},\quad-\frac{\sqrt{1-\sum\xi_{1}^{2}}}{r+\sum x\xi_{1}},$$

but other combinations are possible.

We must choose between these combinations, and secondly, in order to define X1, Y1, Z1 we need a third equation. For such a choice, we must endeavor to bring us closer as much as possible to the law to. Let's see what happens when (always making t = -r ) we neglect the squares of the velocities ξ η etc.. The 4 invariants (5) then become:

$$0,\quad-r-\sum x\xi,\quad-r-\sum x\xi_{1},\quad1$$

and the 4 invariant (7):

$$\sum X_{1}^{2},\quad\sum X_{1}(x+\xi r),\quad\sum X_{1}\left(\xi_{1}-\xi\right),\quad0.$$

But to be able to compare it with the law of, another transformation is needed; here x0 + x, y0 + y, z0 + z are the coordinates of the attracting body at the instant t0 + x, and $$r=\sqrt{\sum x^{2}}$$; in the law of it is necessary to consider the coordinates x0 + x1, y0 + y1, z0 + z1 of the attracting body at the instant t0, and the distance $$r_{1}=\sqrt{\sum x_{1}^{2}}$$.

We can neglect the square of time t required for the propagation and therefore proceed as if the movement was uniform, then we have:

$$x=x_{1}+\xi_{1}t,\quad y=y_{1}+\eta_{1}t,\quad z=z_{1}+\zeta_{1}t,\quad r\left(r-r_{1}\right)=\sum x\xi_{1}t$$