for β rays of radium. We have studied the deviability of these
rays under the influence of an electrostatic field and under that of a
magnetic field produced by electromagnets, that is neutral closed currents,
whose electrons have speeds v' very small in relation to c.� The accelerations always stay small in
these experiments.
We can consequently develop again, in expression (VI) of
the elementary force, the quantities r, u^{2},u_{r} following the formulas of
n^{o} 3, these expansions supposing small not the speeds but the accelerations
of diverse orders. We will have, v being the speed comparable to c,
whereas v'/c is very small,

_{}

_{}

Let v/c=β be a new vector of length comparable to
(Oeuvres 409) unity; we will have

�(35)������� _{}�������

(36)������� _{}

(37)�������� _{}

In
expression (VI) of F_{x} one can develop φ, ψ following Taylor�s formula in the setting of values _{}�since _{}�differ from

these values
only by very small quantities ε, η of the order of _{}, . . . ,we will have

_{}

_{}

(38)������ _{}_{}

(39)������ _{}�

The
expression (VI) of F_{x} will thus become

(40)���� _{}

_{} (Oeuvres 410) The terms in 1/c^{2} will be negligible compared to those of
the first order; for the electrostatic action (v'=0) it will simply become

(41)� _{}

To obtain
the action of an element ds' of a
neutral

closed current
whose electrostatic charge can be neglected, that is for which the positive
and negative charges _{}�are clearly equal by unity of
volume and of opposite signs, we have only to add the sum of the actions of
the positive and negative ions of ds' on the electron e;
let v_{1}' , v_{2}'�
bethe speeds of the positive and negative ions, the conductor
or magnet being at rest, we will have

_{}

In the sum of the actions of both kinds of ions
on e, the terms independent of v' being taken with opposite
signs are canceled in (40), and there remains

(42) _{}_{}

The action
of the current is thus proportional to its intensity. Furthermore, two
elements of parallel current of opposite direction and of the same intensity
have no action. If therefore we consider a uniformly magnetized magnetic
sheet as being a system of very small closed currents of the same intensity,
only the parts of these currents situated on the edge of the sheet will have
a noticeable action; (Oeuvres 411) the effect of
the currents situated at the interior will tend towards zero with the
dimensions and the distances of the currents. Such a magnetic sheet will
therefore be equivalent

�to a closed current circulating on its
contour: this is exactly the reasoning that one makes in electrodynamics, and
it would be easy to give it a more rigorous form.

Similarly, F being a linear function of the
cosine directions _{}�of the current
element, the principle of the sign currents is satisfied for J.

But there ends the analogy with classical
electrodynamics. Thus for weak speeds (that is to say small β) we
saw earlier that the action F_{x}, F_{y}, F_{z} of a
closed current on a moving electron is perpendicular to the velocity of the
latter. It is not so, in general, for β compared tounity.
Indeed, the component R of Fparallel to β will be

�(43)������
_{}�

where we put,
to abbreviate�

_{}

_{}

The quantity Rwill be nil for all
closed circuits only if _{}is a total differential, that is we have

(Oeuvres 412) But β^{2} is
independent of x', y', z' and we have

_{}

_{}.

The
condition necessary and sufficient in order that the force be perpendicular
to the speed will thus be that the expressions f, F, formed by means of φ,ψ, satisfy the differential equation

(46)�������������������������� _{}

For very small β, f is a constant f_{0}
and F is equal to _{}; the relation is satisfied.

When φ andψ do not depend on
β_{ρ}, the relation take the simple form f=0.

Likewise, the action of a closed solenoid, or
of a closed electromagnet, is not nil in general, unless a certain
differential equation of the third order between φ
andψ
is satisfied.

Finally, more generally, the knowledge of the
magnetic field in a point is not sufficient to determine the force that an
electron in very fast motion will experience at that point, unless the
stated relations are satisfied. This last force is determined only when we
give the disposition of currents (assuming the magnets being replaced by the
equivalent currents) and by this the values of β_{ρ} for
the diverse elements of current.

The experiment should thus first of all decide if
the

several laws of electromagnetism just in question apply to the β
rays of radium. Leaving the notion of field, these questions are not even
asked and it is no doubt for this reason that, to my knowledge, no precise
quantitative experiment has yet answered them. Kaufmann�s fine experiments,
undertaken towards a different goal, do not allow answering them, as we are
going to see.

On the whole, the notion of field will apply to
the action that (Oeuvres 413) β rays undergo only under certain
conditions; we have moreover shown that this notion introduces, in general
(that is the forces and the field depending on the speeds), absolute motion.
The ordinary effects of the magnetic field escape this rule only to a certain
approximation (neglecting the motion of the Earth, etc�)