Page:Deux Mémoires de Henri Poincaré.djvu/4

system $x',y',z',t'$ exactly in the same way as we do it in system x, y, z, t. Let us consider, for example, the motion of a point. If, in time dt the coordinates x, y, z undergo the changes dx, dy, dz, then we have for the velocity components

$\xi ={\frac {dx}{dt}},\ \eta ={\frac {dy}{dt}},\ \zeta ={\frac {dz}{dt}}$

However, by these relations the variations dx, dy, dz, dt contain the changes

(2)	$dx'=kl(dx+\epsilon \ dt),\ dy'=l\ dy,\ dz'=l\ dz,\ dt'=kl(dt+\epsilon \ dx)$

of the new variables. It is natural to define the velocity components in the new system by the formulas

(3)	$\xi '={\frac {dx'}{dt'}},\ \eta '={\frac {dy'}{dt'}},\ \zeta '={\frac {dz'}{dt'}}$

which gives us

(4)	$\xi '={\frac {\xi +\epsilon }{1+\epsilon \xi }},\ \eta '={\frac {\eta }{k(1+\epsilon \xi )}},\ \zeta '={\frac {\zeta }{k(1+\epsilon \xi )}}$

To have another example, we can imagine a great number of mobile points whose velocities are continuous functions of the coordinates and time. Let dτ an element of volume located at point x, y, z and let us fix the attention to the points of the system which are in this element at one given moment t. Let $t'_{0}$ be the special value of $t'$ which corresponds to x, y, z, t by the equations (1), and consider for the various points the values of $x',y',z'$ which correspond to the given value $t'=t'_{0}$; in other words, let us consider the positions of the points in the new system, all taken for the same value of "time" $t'$. One might ask after the extension of the element $d\tau '$ of space $x',y',z'$, in which are at this moment $t'_{0}$ the selected points which are in dτ at the time t. A simple calculation, which I omit here, led to the relation

(5)	$d\tau '={\frac {l^{3}}{k}}{\frac {1}{1+\epsilon \xi }}d\tau$

Finally, let us suppose that the points in question carry equal electric charges, and admit that in two systems x, y, z, t and $x',y',z',t'$ we attribute the same numerical values to these charges. If the points are sufficiently close to each other, we obtain a continuous distribution of electricity and it is clear that the charge contained in the element dτ