Page:LangevinStLouis.djvu/20

ether; if we seek, then, what is, in the case of uniform translation, the form that the electron would spontaneously take in order to satisfy the condition of zero variation, we find precisely the oblate ellipsoidal form assumed by Lorentz, with this difference, that the equatorial diameter increases with the velocity instead of remaining constant, as Lorentz considers it; this constancy implies a diminution of the volume as the velocity increases. The equations which express in this case the variation of the longitudinal and transverse mass with the velocity are different from those of Abraham and Lorentz, although giving always an indefinite increase of the two masses as the velocity approaches that of light.

The equations thus obtained for the ratio m/m₀ of the transverse mass m, the only one so far accessible to experiment, to the mass mm/m₀ for very small velocities, as a function of the ratio β=v/V of the velocity of the electron to that of light are:

(1) Invariable spherical electron,

${\frac {m}{m_{0}}}{\frac {3}{4}}\psi (\beta )={\frac {3}{2\beta ^{2}}}\left[{\frac {1+\beta ^{2}}{2\beta }}L{\frac {1+\beta }{1-\beta }}-1\right]$

(2) Variable Electron

${\begin{cases}\mathrm {Equatorialdiameter\ constant} &{\frac {m}{m_{0}}}=(1-\beta ^{2})^{-1/2}\\\\\mathrm {Volume\ constant} &{\frac {m}{m_{0}}}=(1-\beta ^{2})^{-1/3}\end{cases}}$

(28) Comparison. The researches of Kaufmann are not yet exact enough to determine which of these equations represents most nearly the experimental variation of the ratio e/m with the velocity. In order to make the comparison, I have used a process similar to that of Kaufmann, who eliminated the two electric and magnetic fields used to deviate the β rays, seeking to obtain the best concordance possible between the experimental variation of e/m and the theoretical variation calculated on the hypothesis that the mass is entirely electromagnetic.

In order to make this elimination, I draw the two experimental and theoretical curves representing e/m as a function of β, on logarithmic coordinates, and seek for what relative positions of the curves we obtain the best correspondence. The results are given for the three theoretical equations and the same series of experimental values. The experimental points corresponding to four different series are given by Kaufmann, and we see that they correspond equally well with the three theoretical curves.