Let _m₀_ = mass of electron for slow speeds; _m_ = apparent mass of electron at any speed; _u_ = velocity of electron; _V_ = velocity of light. Let β = _u_/_V_; then it can be shown that $$ \frac {m} {m₀} = \frac {3} {4} \psi (\beta) $$ (1), where $$ \psi (\beta) = \frac {1}{\beta^2} (\frac {1 + \beta^2} {2\beta} \log \frac {1 + \beta} {1 − \beta} − 1) $$ (2). The experimental method employed to determine _e_/_m_ and _u_ is similar to the method of crossed spectra. Some strongly active radium was placed at the bottom of a brass box. The rays from this passed between two brass plates insulated and about 1·2 mm. apart. These rays fell on a platinum diaphragm, containing a small tube about 0·2 mm. in diameter, which allowed a narrow bundle of rays to pass. The rays then struck a photographic plate enveloped in a thin layer of aluminium. In the experiments the diaphragm was about 2 cms. from the active material and at the same distance from the photographic plate. When the whole apparatus was placed in a vacuum, a P.D. of from 2000 to 5000 volts could be applied between the plates without a spark. The rays were deflected in their passage through the electric field, and produced what may be termed an electric spectrum on the plate. [Illustration: Fig. 28.] If a magnetic field is superimposed parallel to the electric field by means of an electromagnet, a magnetic spectrum is obtained perpendicular to the electric spectrum. The combination of the two spectra gives rise to a curved line on the plate. The double trace obtained on the photographic plate with reversal of the magnetic field is shown in Fig. 28. Disregarding some small corrections, it can readily be shown that if _y_ and _z_ are the electric and magnetic deviations respectively, _z_ β = κ₁ ----- (3), _y_ and _e_ _z²_ --- = κ ---- (4). _m_ _y_ From these two equations, combined with (1), we obtain $$ \frac {y} {z^2 \psi (\kappa_1 \frac {z} {y})} = \kappa_2 $$ ... (5), where κ, κ₁, κ₂ are constants. Equation (5) gives the curve that should be obtained on the plate according to the electromagnetic theory. This is compared by trial with the actual curve obtained on the plate. In this way Kaufmann[129] found that the value of _e_/_m_ decreased with the speed, showing that, assuming the charge constant, the mass of the electron increased with the speed. The following numbers give some of the preliminary results obtained by this method. Velocity of electron _e_/_m_ 2·36 × 10¹⁰ cms. per 1·31 × 10⁷ sec. 2·48 „ 1·17 × 10⁷ „