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 :—

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

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

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

1. Vide Note 21.