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

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

which gives us

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

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