Page:Deux Mémoires de Henri Poincaré.djvu/7

If one wants to realize the persistence and the balance of an electron while making use of the ordinary notions of mechanics, it is obviously not sufficient to consider the electrodynamic actions. The particle - that we consider here as a sphere carrying a surface charge - would immediately explode because of the mutual repulsions or, which is to same, of the stresses of Maxwell exerted on its surface. Therefore another concept should also be introduced, and Poincaré distinguishes at this place between the "bindings" and the "supplementary forces". He initially supposed that there is only the connection represented by the equation

$r=b\theta ^{m}$

r is the semi-axis of the electron, rθ its equatorial radius, b and m variables that remain constant when r and θ (or one of these quantities) vary with the translation speed v. This granted, we know for any value of v the dimensions of the electron - because we know that $\theta =\left(1-v^{2}\right)^{-{\frac {1}{2}}}$ - and by the ordinary formulas of electromagnetic field, the energy, momentum and the Lagrange function can be calculated. Between these values, considered as functions of v, there must be well known relations. Poincaré shows that they are verified only for $m=-{\tfrac {2}{3}}$, which brings us back to the constancy of volume, that is to say, the hypothesis of Mr. Bucherer and Langevin. But we know already that it is not this hypothesis, but only that of a constant equatorial radius, which is in agreement with the postulate of relativity. It is thus necessary to have recourse to additional forces.

By supposing that they depend on a potential of the form $Ar^{\alpha }\theta ^{\beta }$, where A, α and β are constants, Poincaré finds that the constancy of the equatorial radius requires α = 3, β = 2, i.e. the potential in question must be proportional to volume. It results from it that the sought supplementary forces are equivalent to a pressure or a normal tension exerted on the surface and whose magnitude per unit of area remains constant, whatever the speed of translation. It is immediately seen that only a tension directed towards the interior is appropriate; we will determine the magnitude by the condition of an electron at rest and which has consequently the shape of a sphere, and it must be in equilibrium with the electrostatic repulsions. So when the particle is set into motion, the stress of Poincaré is united with the electrodynamic actions,