Page:CunninghamPrinciple.djvu/5

542 Hence

$$\frac{dW}{dv_{0}}f=+v_{0}K+\frac{v_{0}f}{c^{2}\beta}\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}$$

and therefore

$$f\left\{ \frac{1}{v_{0}}\frac{dW}{dv_{0}}-\frac{1}{c^{2}\beta}\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}\right\} =K$$,

so that the longitudinal mass is equal to

$$\frac{1}{v_{0}}\frac{dW}{dv_{0}}-\frac{1}{c^{2}\beta}\int\rho_{0}x\left(F_{x}\right)_{0}d\tau_{0}$$

It is the second term in this expression that is neglected by Abraham, and which he has to account for by assuming the energy of the electron to be made up of W together with a term not electromagnetic in origin.

We proceed to evaluate this expression in the two cases (i.) of a sphere with uniform volume density; (ii.) of a sphere with uniform surface charge.

(i.) Volume Distribution.

$\begin{array}{l} G=\frac{4}{5}\frac{e^{2}}{ac^{2}}\frac{v_{0}}{\beta};\\ \\W=\frac{3}{5}\frac{e^{2}}{a\beta}\left(1+\frac{v_{0}^{2}}{3c^{2}}\right).\end{array}$|undefined

Hence