We saw that Lorentz’s theory is expressed, in
final analysis, by the statement of an elementary law of action between two
charged elements of volume. A mobile corpuscle P’(x’,y’,z’), carrying
the charge e’, exerts on another P of charge e placed in xyz
a force depending on the speed v of this latter, ofthe direction _{}, . . . and of the radius R of the wave emitted by P’
at the moment t’= t - R/c, and reaching P at instant t;
and finally, of the speed v’ and of the acceleration w’ of P’
in t’. We have (the system of coordinates being at rest in relation to
the ether)

_{},

the equation which defines R as an implicit function of x, y,
z, t when the functions x’=x’(t’), y’,z’ are given. We then conclude

_{}

are
equations analogous to (4) and (5).

The elementary force, of which the expression has been given by
Schwarzschild [1] for the
case of two electrons of

negligible
diameter in relation to their distance, possesses a component parallel to R,
another parallel to v’, and a third to w’, and (Oeuvres 378) we
have

_{}_{}

The equations of the motions are

_{}_{}

the
sum being extended to all electrons and more generally, according to the
principle of d’Alembert,

_{}

We intend, while keeping the equations (V^{a}) and (V), to
modify the expression of F in a manner as to eliminate the absolute
motion. This latter comes to it explicitly

from
the absolute speeds v, v’ and implicitly from the law of propagation.

In the general views expounded previously, the principle of
the impossibility of action at adistance will be expressed by the condition that F
depends only on the disposition and on the speed of particles around e,
in other words, on the vectors U, r and D, as well as the first
derivatives of r, D with respect to x, y, z. These derivatives
introduce, as we already said, acceleration w’. Also, it (Oeuvres 379)
is natural to consider F as proportional to the density D of
particles close to e and to the charge e; we can then, exactly as
in Lorentz’s formula, decompose F in accordance with the directions r,
U, w’ and write

_{}

The quantities A_{1}, B_{1}, C_{1},
which are independent of the system of coordinates, depend only, by hypothesis,
on r, U^{2}, U_{r}. SinceU_{x}=c cosrx-u_{x} and U^{2},
U_{r} are expressed linearly byu^{2}, u_{r}, we can write also

_{}

where
A_{2}, B_{2}, C_{2} are functions of u^{2},
u_{r}; and we will suppose them independent of r. Moreover,
for these expressions containing the speeds c, u in an homogenous form,
we can write, for the speeds thatare relatively small in relation to that of light (n,
m, p being exponents carefully chosen)

Although this hypothesis is not indispensable, we will
suppose that if we change the sign of the speeds, A_{2}, B_{2},
C_{2} will not change. In other words, they are even functions of _{}; the expansions will not contain odd powers of _{}.

By replacing D with its value from (5), and disposing of
some of the coefficients _{}we will finally be
able to write F_{x} in the (Oeuvres 380) form

_{}

where
k, a_{i}, b_{i}, c_{i}are coefficients
remaining arbitrary in the present state of theexperience, which
also implies thatφ, ψ, χ are almost unknown functions.

The terms of an order superior to the second play a role
only in the pressure of light (which is in the domain of optics, consequently
they will not concern us) and in the study of the β rays of radium. It is
not surprising, then, that they are not well determined. Yet, we will see that
even the terms of the second order, on which the electrodynamic phenomena
depend, are not entirely determined by the experiment:the quantity k remains

arbitrary.
The proposed formula is therefore sufficiently general for the goal we are
seeking. It is not the most general one, and considerations of the rotary
[orbital?]movements ofelectrons would be less general for
different reasons.

The equations of motion will still be (V) and (V^{a}).
In the previously discussed case when waves emitted at different instants by
the same electron e’ simultaneously reach the electron e, we will
have the sums _{}to take account each of these actions.

To show that the new theory effectively takes account of all the
facts known in the field of electrodynamics, I will consider first the case
where speeds and accelerations are relatively small. It is so, particularly,
for all the phenomena belonging to the field of classical electrodynamics, the
cathode rays, etc. This case is characterized by (Oeuvres 381) the fact that we
can develop all functions of the form _{}playing a role in it,
in very convergent series, proceeding according to the powers of _{}, and taking only the first terms. This is what I call slow
variation phenomena.

Secondly, I will consider the case of just about any acceleration
with very small speeds in relation to c: this is the case of hertzian
oscillations and - up to a certain point - of optics.

Finally, when speeds are comparable to c, but the
accelerations remain small, we are led to Mr. Kaufmann’s experiment on β
rays of radium.

All the electric phenomena observed until now belong to one of
these categories.

We know that the law of the conservation of energy in its
classical form W=const is not applicable any more when there is radiation; we
must resort to the image of a

fluid energy or consider the vibratory
energy lost by a luminous body or by a hertzian exciter as projected in space
with the speed of light and attributing to it a momentum according to Poincaré’s theorem. From here to the ideas we
just expressed there is but one step. But by going into this subject, I would
go beyond the purely critical aim I set for myself. I will therefore confine
myself to demonstrate that forces, motions and consequently, the work are
exactly the same as indicated by Maxwell’s theory, in all the phenomena
observed until now; the law of energy, in the measure that it was verified
inthis
domain, is the consequence

With the principle of energy,
Maupertuis’ principle of least action, which implies the equation of energy in
its classical form, ceases to be true in general, likewise the canonical
equations and the Hamilton-Jacobi equation of partial derivatives. Nothing lets
us believe that Hamilton’s principle will escape from the general rule. In
order to be applicable to formula (VI), it would be necessary to introduce in
it a term containing the acceleration of e, in other words, an inertial
reaction depending on the disposition of external charges. (Oeuvres 382)
Weber’s formula contained a term of this sort and Helmholtz [2]
demonstrated the inadmissible consequences resulting from it. For an isolated
electron, for which _{}approximately, and which is mobile inside an empty charged
sphere, everything happens as if the mass were diminished proportionally to the
potential of the sphere and were equal to zero for _{}volt (positive)
approximately. The acceleration would then become infinite and we can see that
the experiment would not be too difficult to execute.

In
optics, and for the spectral vibrations where inertial reaction plays an
essential role, the influence of electrical charges from the apparatus on the
optical properties and the position of rays in the spectrum resulting from the
introduction of such a term would be absolutely contrary to the experiment. It
is only when Lagrange’s function depends linearly on the speed v
of e, as in Lorentz’s theory [3],
that these consequences are avoided: but all this demands the introduction of
absolute motion. Hamilton’s principle should then beequally
transformed when we consider only relative speeds.

The equations of motion we just wrote are of the second order like
those of mechanics, but because of the argument _{}, they are, at the same time, functional equations and this
very complicated, mixed form cannot be definitive.

[1]Göttinger Nachrichten, 1903, p.
132 and following. The formula includes some restrictions without importance
for our purposes.

[2] H. von Helmholtz, Wissenschaftliche
Abhandlungen, vol. I, Leipzig, 1882, pp. 553, 636, 656.

[3]See the formula (XVII) by
Schwarzschild in the first Part.