Page:PoincareDynamiqueJuillet.djvu/46

by -rξ₁, which gives:

$X_{1}=\alpha \left(x+\xi _{1}r\right)=\alpha x_{1}$

Newton's law would give:

$X_{1}=-{\frac {x_{1}}{r_{1}^{3}}}.$

We must therefore choose, for the invariant α, one that reduces to $-{\tfrac {1}{r_{1}^{3}}}$ to the order of approximation adopted, that is to say ${\tfrac {1}{B^{3}}}$. The equations (9) become:

(11)

${\begin{cases}X_{1}={\frac {x}{k_{0}B^{3}}}-\xi _{1}{\frac {k_{1}}{k_{0}}}{\frac {A}{B^{3}C}},\\\\Y_{1}={\frac {y}{k_{0}B^{3}}}-\eta _{1}{\frac {k_{1}}{k_{0}}}{\frac {A}{B^{3}C}},\\\\Z_{1}={\frac {z}{k_{0}B^{3}}}-\zeta _{1}{\frac {k_{1}}{k_{0}}}{\frac {A}{B^{3}C}},\\\\T_{1}=-{\frac {r}{k_{0}B^{3}}}-{\frac {k_{1}}{k_{0}}}{\frac {A}{B^{3}C}}.\end{cases}}$

We first see that the corrected attraction is composed of two components, one parallel to the vector joining the positions of the two bodies, the other parallel to the velocity of the attracting body.

Recall that when we talk about the position or velocity of the attracting body, it is its position or its velocity when the gravitational wave leaves; for the body attracted, on the contrary, it is its position or its velocity when the gravitational wave reaches it, the wave is assumed to propagate with the speed of light.

I think it would be premature to push further discussion of these formulas, I will confine myself to a few remarks.

1° The solutions (11) are not unique; we can indeed replace ${\frac {1}{B^{3}}}$ which enters in the factor everywhere, by

${\frac {1}{B^{3}}}+(C-1)f_{1}(A,B,C)+(A-B)^{2}f_{2}(A,B,C)$,

f₁ and f₂ are arbitrary functions of A, B, C; or we are taking β no longer as zero but adding arbitrary complementary terms to α β γ, provided they satisfy the condition (10) and are of the 2^nd order with regard to ξ as far as α is concerned, and of the 1^st order as far as β and γ are concerned.

2° The first equation (11) can be written:

(11bis)

$X_{1}{\frac {k_{1}}{B^{3}C}}\left[x\left(1-\sum \xi \xi _{1}\right)+\xi _{1}\left(r+\sum x\xi \right)\right]$

and the quantity in brackets can, itself, written as:

(12)	$\left(x+r\xi _{1}\right)+\eta \left(\xi _{1}y-x\eta _{1}\right)+\zeta \left(\xi _{1}z-x\zeta _{1}\right),$