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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 260 (Oeuvres 413)

      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=βc 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) Göttinger 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 βy= βz=0,  βx= β,  βρ= βcos(ρ,x). Let σ be the density of the electricity on the plates. The components of the force exerted on the electron will be, according to formula (41),


(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 y'=0 (first plate), then for y'=a (second plate); it is necessary to get the difference between the results. Let us have y' undetermined for the moment. As we have admitted in section 2 that φ and ψ are even functions of βρ, the function to be integrated has, in the first and third integral, opposite values for points (x-x'), (z-z') and -(x-x'), -(z-z'); these integrals are therefore nil: the force is parallel to y by symmetry.

    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, z'=0, integrating by relation to ρ from to ρ=¥, then from to; the integral sought for will be four times the result obtained.

     We have


     For the upper limit ρ=¥, this becomes  for y-y' >0 (first plate, y'=0) and for y-y' <0 (second plate, y'=a). The lower limit gives arctan=0; we have therefore for the whole of the two quarters of the plate of the condenser


     (Oeuvres 415) Let us denote; we finally have for the total force sought


E being the electrostatic field in the ordinary sense. In our theory this field does not exert, as in Lorentz’s theory, a force eE, 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', dy to –dy'; the actions of corresponding elements are canceled in the first integral Rmy, the latter is therefore nil by symmetry. On the other hand, the changes in speed that the forces produce being small in relation to the initial speed in this experiment [], an action parallel to the latter, which does not consequently directly produce a deviation, is negligible at first approximation; the magnetic action observed will therefore be here perpendicular to the field and to the speed, as Lorentz’s theory would have it and as the experiment shows. We see that in these conditions, (Oeuvres 416) the general problems raised earlier are not resolved by these experiments.

     In his definitive research, Kaufmann used permanent magnets in the shape of a horseshoe; we cannot calculate Rmz 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 Rey and Rmy 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=m0Φ(β') is the mass function of the speed and reduces itself to m0 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)


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.=m0 , Lorentz’s theory gives the parabola


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.


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