Page:PoincareDynamiqueJuillet.djvu/40

We have also:

$x+\epsilon t=x-\xi t,\quad r^{\prime 2}=k^{2}(x-\xi t)^{2}+y^{2}+z^{2}$

and

(4)	$X_{1}={\frac {-k(x-\xi t)}{r^{\prime 3}}},\quad Y_{1}={\frac {-y}{kr^{\prime 3}}},\quad Z_{1}={\frac {-z}{kr^{\prime 3}}};$

which can be written:

(4bis)

$X_{1}={\frac {dV}{dx}},\quad Y_{1}={\frac {dV}{dy}},\quad Z_{1}={\frac {dV}{dz}};\quad V={\frac {1}{kr^{\prime }}}.$

It seems at first sight that the indetermination remains, since we have made no hypothesis about the value of t, that is to say about the speed of transmission; and that also x is a function of t, but it is easy to see that x - ξt, y, z (which appear in our formulas) do not depend on t.

We see that if two bodies are simply in motion by a common translation, the force acting on the body is drawn normal to an ellipsoid with its center at the attracting body.

To go further we must look for the invariants of the Lorentz group.

We know that the substitutions of this group (assuming l = 1) are linear substitutions which do not affect the quadratic form

$x^{2}+y^{2}+z^{2}-t^{2}.\,$

Let on the other hand:

$\xi ={\frac {\delta x}{\delta t}},\quad \eta ={\frac {\delta y}{\delta t}},\quad \zeta ={\frac {\delta z}{\delta t}};$
$\xi _{1}={\frac {\delta _{1}x}{\delta _{1}t}},\quad \eta _{1}={\frac {\delta _{1}y}{\delta _{1}t}},\quad \zeta _{1}={\frac {\delta _{1}z}{\delta _{1}t}};$

we see that the Lorentz transformation will cause to make δx, δy, δz and δ₁x, δ₁y, δ₁z, δ₁t undergo the same linear substitutions as with x, y, z, t.

We regard

${\begin{array}{ccccccc}x,&&y,&&z,&&t{\sqrt {-1}},\\\\\delta x,&&\delta y,&&\delta z,&&\delta t{\sqrt {-1}},\\\\\delta _{1}x,&&\delta _{1}y,&&\delta _{1}z,&&\delta _{1}t{\sqrt {-1}},\end{array}}$

as the coordinates of three points P, P', P" in a 4-dimensional space. We see that the Lorentz transformation is a rotation of that space around the origin, regarded as fixed. We shall therefore have no other distinct invariants than 6 distances of the 3 points P, P', P" between them and the origin, or, if you like