Page:SahaElectrodynamics.djvu/32

 Let us now consider, following the usual method of treatment, the longitudinal and transversal mass of a moving electron. We write the equations (A) in the form

$$\left. \begin{array}{c} m\beta^{3}\frac{d^{2}x}{dt^{2}}=eX=eX'\\ \\m\beta^{2}\frac{d^{2}y}{dt^{2}}=e\beta\left[Y-\frac{v}{c}N\right]=eY'\\ \\m\beta^{2}\frac{d^{2}z}{dt^{2}}=e\beta\left[Z+\frac{v}{c}M\right]=eZ'\end{array} \right\}$$

and let us first remark, that $$eX', eY', eZ'$$ are the components of the ponderomotive force acting upon the electron, and are considered in a moving system which, at this moment, moves with a velocity which is equal to that of the electron. This force can, for example, be measured by means of a spring-balance which is at rest in this last system. If we briefly call this force as "the force acting upon the electron," and maintain the equation :—

$$\text{Mass-number} \times \text{acceleration-number} = \text{force-number}$$, and if we further fix that the accelerations are measured in the stationary system K, then from the above equations, we obtain :—

$$\text{Longitudinal mass} = \frac{m}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{\frac{3}{2}}}$$

$$\text{Transversal mass} = \frac{m}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

Naturally, when other definitions are given of the force and the acceleration, other numbers are obtained for the