have lead to the belief that the mass of corpuscles or electrons depends
on their velocity and is entirely of electromagnetic origin. The existence
of an electric inertial reaction and its variability with velocity was
anticipated by the theory which at a first glance seems to have received
a remarkable confirmation. However in view of the scope of these
conclusions it is advisable to find out if they are absolutely
indispensable.
Let's recall that in these experiments a beam of β rays is
simultaneously subjected to the action of an electric field E, producing a
deviation y, and to a magnetic field H parallel to E, producing a
deviation z perpendicular to the first. A photographic plate,
perpendicular to the non-deviated rays, records an impression of the rays,
and permits a direct measurement of y and (Oeuvres 355) of
z. With m and v the mass and velocity of an
electron, e its charge, and a and b two constants
of the apparatus. We have, according to Lorentz's theory,

Radium, within certain limits, emits rays of all velocities. These
equations, where v plays a parametric role, represent a certain
parabola if m doesn't depend on v. But the curve
observed by Kaufmann is different. This occurrence calls for various
explanations.
1° According to Lorentz,
the movement of an electrified body with respect to the ether is
equivalent to an electric current whose field, by an effect analogous to
self induction, reacts on the body and produces a force which, in the
most general hypotheses, is a linear function of the acceleration
components, the coefficients (for transverse and longitudinal mass)
being functions determined by the absolute velocity v known,
for example, for the sphere and ellipsoid. In the case of Kaufmann's

experiments, only the transverse mass comes into play. If we use the function
in (16)
instead of m we very closely approximate the observed curve,
whether we consider the electron as a rigid sphere (Abraham) or
whether we only consider its volume to be constant (Bucherer and
Langevin).
To appreciate the value of this interpretation let's recall that the
calculation of electrodynamic mass relies uniquely on the motion of
the electrified body with respect to the ether, the position and
velocity of other bodies being of no consequence. It is the body's
absolute velocity which is used in the formula Under this
interpretation Kaufmann's experiment will be therefore the first to
present evidence of absolute motion. Now, regarding this delicate
question, Lorentz's theory, at least in the form as expressed in
section 1, is in disaccord with the experiment, and this disaccord
carries over in particular to the expression of electromagnetic
momentum G (1), from which Abraham (Oeuvres 356) has deduced
values for longitudinal and transverse masses. Indeed, from the
calculation of G we deduce(2), for the case of a charged condenser
carried along by the earth's translational motion, the existence of a
second order couple impressed on the condenser. But Trouton and
Noble, who did this experiment(3), didn't observe the couple. Therefore
the quantity G doesn't depend on the absolute velocity or, at least not
in the same manner as in Lorentz's theory. We have to conclude that,
even if the agreement between Abraham's theory and Kaufmann's
experiments were perfect, this theory should nonetheless be considered
as doubtful.

(1) The vector is, as we know, the integral of the radiant vector
extended over all space, multiplied by .
(2) H.A. LORENTZ, Elektronentheorie, p. 257.
(3) London Trans., A. t. CCII, 1903, p. 165.

2° In striving to eliminate
the influence of absolute motion in his equations Lorentz was led to
certain new hypotheses, to which I will return in the next section.
The dimensions of electrons, in particular, would be reduced to
of their value when animated by the absolute velocity p. This
hypothesis leads to new formulas for mass that Kaufmann considered as
irreconcilable with his last experiments.(1) But this conclusion seems
doubtful to me. Indeed, let's take for H, E, a, b, the
directly observed values, where instead of the

value which corresponds to cathode rays, let's take, in the formula of
Langevin and Bucherer

in that of Abraham

(Oeuvres 357)

and finally in that of Lorentz

.

This means to multiply in the ratios of 1 to 1.040, 1.070 and 1.130, the
abscissas and the ordinates calculated by Kaufmann for
(loc cit., p. 534). We thus find three curves for which the
errors are on the order of the experimental errors, as we verified in
displaying

(1) Annalen der Physik., t. XIX, 1906, p. 487; see also
M. PLANCK, Physik. Zeitschr., t. VII, 1906, p. 754.

these values on the curve, fig. 11, of Kaufmann. That savant has
done direct observations and then using the method of least squares
deduced from the various proposed formulas a certain constant
; independent
of the hypotheses developed on
; Lorentz's
formula giving an inadmissible value. Our calculation shows that the
value of ,
determined in this manner, involves a considerable uncertainty, for
we took the directly observed value, and the errors that result don't
exceed the admissible magnitude. As to the value
its application to the &beta rays of radium can't be permitted, since
it is not, in general, applicable for the Zeeman effect.
In short, these observations don't therefore permit the preference of
any one of these formulas over another, and it would be easy to
expand on this more so.
But it is important to observe that Lorentz's new hypotheses involve
a modification in the expression for the force that two moving
electrified bodies exert on one another, a modification which, as
shown in a simple argument, is only appreciable for velocities
comparable to that of light. That is to say, for Kaufmann's experiment
only.
3° This leads to a general
remark. It is easy to introduce the terms presenting this peculiarity into
the equations of electrodynamics. Since the system of equations (I) to
(IV) can be replaced by elementary actions, it suffices to consider them
here. Now the force exerted by the particle e', having velocity
v', (Oeuvres 358) on the particle e, having velocity
v, is expressed in this latter approach in a linear form with the
factor . It is
expressed in the former for the case of uniform motion, in a very
complicated form

(given by Schwarzschild). We saw, in section 6, the first terms of
the expansion for when v'/c is small. The dissymmetry
established thusly between v and v' is not confirmed by
experiment, and there are, as we will see in the Second Part, an infinity
of formulas containing only relative velocities and consequently
differing from that of Lorentz by the terms in
. For a
better reason, we can add the terms containing v/c, of the
third and fourth orders, without which the formulas fail to be in accord
with all the experiments when v/c is small. Lorentz's
elementary actions formula can be only the beginning of a serial
development. We will be able to dispose of the arbitrary function
v thus introduced in (16) in order to produce agreement with
Kaufmann's experiments by the hypothesis of constant mass,
and in a manner so as to completely safeguard the relativity of motion.
This is what will be shown in greater detail in the Second Part of this
work.
But there is more. The very form of the curve, and the existence of a
point, where the deviations y and z are null, and the
velocity equal to c, remain doubtful. Indeed, let's suppose

from whence [we get] the parabola

whereas, according to Lorentz, for m = const. we have the

The values of E, H, a, b, being those given by Kaufmann, and
,
and for the refined values of y', z', Kaufmann
(loc. cit., p. 529), the parabola becomes

The difference between the various individual values of y'
observed by Kaufmann and the curve which represents the average for
his experiments is often greater than 0.0030. The third column errors,
although systematic, must therefore be considered as admissible.
Those in the fourth column, which correspond to the hypothesis

are completely explainable as experimental errors, except for the last
two. But we must note that a one percent error in the absolute
measurement of the magnetic field, which can very well result from
an accumulation of errors that accompany the various observations
necessary for this experiment, would be especially sensible in these
two instances, and change their y' by two percent, that is

to say, by 0.0034. These last points have been observed only two times
and Kaufmann observes that we are inclined, at the extremities of the
curve where the intensity is weak, to scrutinize the extension, that is
to say the tangent, where there results a value too weak for y'.
These two considerations would be sufficient to explain the errors in
the fourth column. But, in this curve for (17), the critical aspect is
not the velocity of light c, but the maximum relative
velocity (Oeuvres 360) 2c of two light rays within the
same fixed system, for which there is nothing surprising in a theory
which considers only relative velocities.
The magnetic deviation z' is nil at this point, but not the
electric deviation, which is almost equal to half the width (0.03) of
the curve.
We see how expansive the field is which remains for hypotheses.
In closing, let's note that the velocity v simply performs a
parametric role, and is determined from each point on the curve by
means of observed values of y and z. The result
differs according to the theory used, and we can represent a curve in
an infinite number of ways for a given parameter. It would be
otherwise if direct and precise experiments, such as the ones executed
by Des Coudres and Wiechert through the means of Hertzian
oscillations on cathode rays, could give a direct determination of
v, but such experiments don't seem realizable.
Kaufmann's experiments can equally be interpreted by modifying
the existing laws of electrodynamics in a manner that eliminates
absolute motion and by making the electrodynamic mass constant.
We can no longer conclude from these experiments that the mass of
electrons is of electromagnetic origin; but it remains possible and the
unity of