Page:Electromagnetic phenomena.djvu/19

published by Kaufmann^[1] in 1902. From each series he has deduced two quantities η and ζ, the "reduced" electric and magnetic deflexions, which are related as follows to the ratio $\beta ={\frac {w}{c}}$:

$\beta =k_{1}{\frac {\zeta }{\eta }},\psi (\beta )={\frac {\eta }{k_{2}\zeta ^{2}}}$.

(34)

Here ψ(β) is such a function, that the transverse mass is given by

$m_{2}={\frac {3}{4}}\cdot {\frac {e^{2}}{6\pi c^{2}R}}\psi (\beta )$

(35)

whereas k₁ and k₂ are constant in each series.

It appears from the second of the formulae (30) that my theory leads likewise to an equation of the form (35); only Abraham's function ψ(β) must be replaced by

${\frac {4}{3}}k={\frac {4}{3}}\left(1-\beta ^{2}\right)^{-1/2}$.

Hence, my theory requires that, if we substitute this value for ψ(β) in (34), these equations shall still hold. Of course, in seeking to obtain a good agreement, we shall be justified in giving to k₁ and k₂ other values than those of Kaufmann, and in taking for every measurement a proper value of the velocity w, or of the ratio β. Writing $sk_{1},\ {\frac {3}{4}}k_{2}^{'}$ and β' for the new values, we may put (34) in the form

$\beta '=sk_{1}{\frac {\zeta }{\eta }}$

(36)

and

$\left(1-\beta ^{'2}\right)^{-1/2}={\frac {\eta }{k',\zeta ^{2}}}$

(37)

Kaufmann has tested his equations by choosing for k₁ such a value that, calculating β and k₂ by means of (34), he got values for this latter number that remained constant in each series as well as might be. This constancy was the proof of a sufficient agreement.

I have followed a similar method, using however some of the numbers calculated by Kaufmann. I have computed for each measurement the value of the expression

$k_{2}^{'}=\left(1-\beta ^{'2}\right)^{-1/2}\psi (\beta )k_{2}$

(38)

that may be got from (37) combined with the second of the equations (34). The values of ψ(β) and k₂, have been taken from Kaufmann's tables and for β' I have substituted the value he has found for β, multiplied by s, the latter coefficient being chosen with a view to

1. Kaufmann, Physik. Zeitschr. 4 (1902), p. 55.