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Translated (1980) from Recherches critiques sur l'Ėlectrodynamique Générale,
Annales de Chimie et de Physique, Vol. 13, p. 145, 1908.

Annales 181 (Oeuvres 347)

     With the ether acting on ions without undergoing action itself, Newton’s principle is not satisfied by Lorentz’s theory, and Poincaré1 has shown that we have for the resultant of translation

where the integrals are extended over all space and S is the radiant vector.  Furthermore, an electrified body in uniform motion exerts on itself, in general, a couple.  It is important to consider separately the diverse aspects of the question which this poses: can we, from the view-point of the facts, draw from this inequality of action and reaction an objection to Lorentz’s theory?  The answer is affirmative. (Oeuvres 348)

     Lets consider, at first, two electrons with charges e,e’, with coordinates x,y,z; x’,y’,z’, velocities v,v’ and accelerations


Annales 182


w,w’, placed a great distance apart relative to their dimensions. Liénard2 and Wiechert3 have shown that for the potentials produced by e’ we have

where we have to take the quantities in brackets at a previous instant t – r/c such that the wave emitted at this instant reaches (xyz) at t; the vector r is directed from e’ towards e, and we have the equation

     It will suffice to consider the particular case where the velocities and accelerations are small, so that we can set

     A simple calculation, that we will furthermore find in the Second Part, leads to the development of θ and A, from which comes, for the force Fx exerted by e’ on e, the expression

where all the quantities v’, w’, v must be taken at

Annales 183

the instant t and where r is the actual distance between the points e, e’.

     This expression contains velocities and accelerations in a nonsymmetrical manner that clearly shows the inequality of action and reaction, even when the accelerations are supposedly negligible and there is no radiation.  In the (Oeuvres 349) case of uniform translation motion of the points we have

v = v’,  w’ = 0;

the term which is multiplied by  cos (r,x) is directed along the line of junction ee’ and satisfies the principle; the term gives a force parallel to v, applied to e, and another one, equal and opposite, applied to e’.  If the charges e, e’ were connected by a rigid member, these two forces would produce a couple whose axis would be perpendicular to the velocity v and the junction line r.

     In the Second Part we will see that no experiment requires this dissymmetry where the velocities are concerned, and that it is evident a priori.  Since no experiment has shown anything but relative motions, expression (13) must be replaceable by another, of the second degree, which contains relative velocities only.  Such an expression, constrained to be a vector component, wouldn’t present such a dissymmetry.

     On the other hand, one experiment by Trouton and Noble4, which should have, for the case of a charged condenser, shown evidence of the couple under question, gave a negative result.  In that it concerns the terms relative to velocities, the inequality of action and reaction constitutes therefore a serious objection to Lorentz’s theory.

     We can’t say very much, from an experimental point of view

Annales 184

about the nonsymmetrical  terms which are dependent on the acceleration w’.  They contribute, even at small velocities, and when certain conditions of symmetry are satisfied, to electromagnetic mass and more generally to a reaction of inertia.  For a uniformly charged sphere, or radius R, the result of elementary actions


the quantity  is therefore the electromagnetic mass and even in (Oeuvres 350) making the same abstraction from* Kaufmann’s experiments nothing permits denying the possibility of such a reaction.  On the contrary, it is evident that there’s a considerable advantage, from the point of view of unifying our concepts, to be able to deduce the reaction of inertia and kinetic energy from electromagnetic energy.  We will further study the question of the variability of mass as a function of velocity.

*The phrase “en faisant même abstraction” was rendered as “in excluding” in the 1980 hard copy.

     Hertz’s theory satisfies the principle [of action and reaction?] in a general manner.  For example,  with the pressure that light exerts on a body immersed in dielectric air or ether there corresponds a reaction of the same magnitude applied to these dielectrics5, in such a way that, in the first case, the principle is satisfied by considering the medium only.  But experiment has shown the existence of this pressure, even in the most perfect vacuum. In this latter case there is no reaction according to Lorentz, but, according to Hertz, there really is one, and the ether is set into motion.  However, to make this [reaction] perceptible, the ether would have to quit


Annales 185

concealing itself in all the experiments.  Since it doesn’t respect this wish, it is difficult to say if, in this case, whether the logical advantage is on Lorentz’s side, who simply expresses the idea of action without reaction, or whether it is on Hertz’s side who saves the principle, but in such a manner that it becomes a simple agreement.

     If we are content with the forces exerted by ions on one another existing without the intervention of an intermediate, such as ether, then the finite speed of propagation leads to the lack of simultaneity and to the inequality of actions of ions on one another when they are separated (generally at least).

     In the classical theories of Optics, for example in Sellmeier and Helmholtz’s dispersion theory, the action of light on molecules is equal to the reaction of the molecules on the ether. The principle was never considered as being applicable solely to the material.  What we can object to, in the theory, is that it would be more satisfactory if the intermediate were devised in such a manner as to explain the matter of the equality of action and reaction, and I indicated in the Introduction that radiant energy materializing and projecting at the speed of light constituted such an (Oeuvres 351)  intermediate.6 We return therefore, in a new form, to the emission theory, and to use Poincaré’s example, the recoil of an artillery piece and the force experienced by a body which transmits a wave of radiant energy in a certain direction are absolutely analogous, which is not the case when, instead of using this model, we consider energy to be propagated (the ether theory).

     Poincaré has shown that the inequality of action and

Annales 186

reaction doesn’t lead to perpetual motion in Lorentz’s theory; additionally, under these conditions, we are obliged to admit the hypothesis of retarded potentials.



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1 Archives néerl., 2nd series, vol.5, 1900, p. 252: Électricite et Optique, p. 448.

2 LIENARD, L’Éclairage électrique, vol. 16 1898, p. 5, 53, 106.

3 WIECHERT, Archives néerl., 2nd series, vol. 5, 1900, p. 549

4 London Transact., A, vol. 202, p. 165.

5 POINCARÉ, loc cit.

6 It also permits avoiding absolute motion and the other difficulties (see Introduction and Second Part).