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WALTER RITZ

We know that in his experiments, Kaufmann observed
the deviation of a beam of β-rays, directed parallel to the *x*-axis,
by an electric field parallel to the *y*-axis, produced by a plane
condenser and a magnetic field in the same direction,** **produced by an
electromagnet or a permanent magnet. We observed the deviations *y, z*,
mutually perpendicular, produced by the combined action of the two fields;
these deviations depend on the speed *v=*β*c* of the electron in
motion or β-ray, and this is the curve *y=y*(β), *z=z*(β)
(where β is not directly known, but simply plays the role of a parameter)
that the experiment gives. The apparatuses were furthermore symmetrical in
relation to the

_________________________________________________________________________

(^{1}) *G**ö**ttinger Nachrichten, *1901, v.1; 1902, v.5; 1903, v.3. *Annalen d. Physik,*
v. XIX, 1906, p.487.

Annales 261

*y*-axis,
the fields sensibly homogeneous. We are going to see what the new theory gives
us under these conditions.

*Electrostatic action*. – The plates of a
condenser being parallel to the *x-z* plane, let *y* and *a-y*
be the distances of the moving electron from these two plates. We obviously
have β* _{y}*= β

_{}

(Oeuvres 414)

It is permissible to consider, at first
approximation, the extent of the condenser to be very great; the integrals must
therefore be extended throughout the whole *x '-z'* plane
and taken at first for

Let _{} be the angle that ρ makes with β,
that is with the *x*-axis; we have

_{}

Expressed by means of the
variables ρ and_{}, the
second

Annales 262

integral becomes

** **[Translation editor’s note**. _{}**was
omitted in original text.]

_{}

We will obtain the action of the quarter-plane
included between lines *x '*=0,

We have

_{}.

For the upper limit ρ=¥, this becomes** **_{} for *y-y ' *>0 (first plate,

_{}.

(Oeuvres 415) Let us denote_{}; we
finally have for the total force sought

(47)
_{}

E being the electrostatic field in the ordinary sense. *In our theory
this field does not exert, as in Lorentz’s theory, a force e*E*, but a
force depending on the speed*

_{}

Annales 263

*Magnetic action*. – We have

_{}

The integrals are extended to all the currents,
including those that in the ordinary sense are equivalent to active magnetic
masses. The functions φ, ψ depend only on _{} and are even functions of this argument. The
electron is sensibly propelled along the *x*-axis (*y*=0, *z*=0),
and there is symmetry in relation to the *y*-axis. Let us therefore change
*y* to –*y '*,

In his definitive research, Kaufmann used permanent
magnets in the shape of a horseshoe; we cannot calculate R* _{mz}* without knowing the distribution of the magnetism.
It is sufficient, for the goal that I have set myself here,

Annales 264

to state that forces R_{ey}_{ }and R_{my}_{ }are
functions of β, the first depending only on the function φ, the
second, moreover, on ψ; these two functions are arbitrary, with respect to
the first terms of their expansion for very small β. They produce the
deviations *y=f*(β), *z*=F(β) and it is clear that one may,
by a suitable choice of φ and ψ, represent the arc of the curve
observed by Kaufmann, all the more so since, as it results from the research of
the scientist, his experiments do not precisely allow the determination of the
coefficients of the first terms of the developments in β. Lorentz’s theory
gives

_{}

where A and** **B
are constant, β*'* designates the ratio of
the [electron’s] speed to *c*, and *m=m*_{0}Φ(β*'*) is the mass function of the speed and reduces itself
to *m*_{0} for β*'**=*0. Now one can always put our solution *y=f*(β),
*z=F*(β) in this new form; it suffices to put

_{}

from where

_{}

that is to say to
introduce a new parameter β*'* and a function Φ of this
parameter, instead of β coming back to the form of Lorentz. In both of
these theories, everything happens therefore as if the mass was a function of
the speed, the values of the latter deduced from the two theories being the
only differences. A direct measure of the speed, such as that executed by
Wiechert for cathode rays, using hertzian oscillations seems scarcely possible.
We conclude from this what we wished to demonstrate:

Annales 265

*Kaufmann’s
experiments are explained either by admitting *(Oeuvres 417)* absolute motion with variability of the mass, or by
considering masses as constant and motions as relative, and admitting that, for
great speeds, the electrodynamic forces are no longer simple linear functions
of the speed, as Lorentz’s theory would have it, but take a more complicated
form.*

In the First
Part, I said** **that, for uniform motions, the action of an electron *e'*
on *e* is a complicated function of its speed *v'* in Lorentz’s
theory, and that nothing permits the admission of such a dissymmetry in *v*
and *v'*.

It is
interesting to calculate the curve obtained when β is small. We content
ourselves in (VI) with the terms of the second order, used in electrodynamics,
that is formula (13). We then have

_{}

_{} (H = magnetic
field)

(50) _{}

from where

_{}

A and B being
constants already defined, *k* therefore disappears from the result. Let
[]

_{}

the mobile
electron will seem to have a variable mass

_{}

Annales 266

The formula
that Lorentz(^{2}) obtained,
while trying to eliminate (Oeuvres 418) absolute motion
from his equations, is

_{}

The first
terms of the two formulas therefore coincide.

For *m*=const.=*m*_{0} ,
Lorentz’s theory gives the parabola

(51) _{}

that is

_{}

It is
noteworthy that the parabola for which there is concordance with observation,
as I have shown in the first part (section 9), would be obtained by replacing _{}in
(50) by _{}, and
that the parabolas (50) and (51) are consequently at equal distances, as
measured on the *y*-axis, on all sides of the observed curve.

In general manner, one can expect, in a theory based on the principle
of relativity, that speeds equal to or greater than the speed of light present
peculiarities as strange as in Lorentz’s theory. Relative speeds much greater
than *c* will have to be taken into consideration for the mutual action of
two β-rays emitted in opposite directions by a grain of radium, and *c*
could not in any way be a critical speed, nor β=1 a singular point on the
curve.

(^{2}) *Amsterdam** Proceedings*,
1904.

Annales 267

As we have
seen earlier, nothing in our theory prevents the admission that the inertial**
**reaction of electrons is entirely electromagnetic in origin. If the
particles ejected by radium are not spherical, the inertial reaction depends on
their orientation: the same molecular
force will communicate different speeds to differently oriented particles, and
the exterior field will give different deviations; if the mass is determined by
one single parameter, as is the case for the ellipsoid of revolution, it will
seen that the mass is a well determined function of the initial speed of the
particle in relation to the radium.

(Oeuvres 414) We
see how any conclusion one way or another would be premature in the not yet
well-explored domain of great speeds.