Page:CunninghamPrinciple.djvu/6

$\therefore$

${\begin{array}{rl}{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&-{\frac {1}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}\\\\&={\frac {1}{5}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}-1+{\frac {v_{0}^{2}}{c^{2}}}\right\}\\\\&={\frac {4}{5}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\\\\&={\frac {dG}{dv_{0}}}.\end{array}}$.

(ii.) Surface Distribution.

${\begin{array}{rl}G&={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}{\frac {v_{0}}{\beta }}.\\\\W&={\frac {1}{2}}{\frac {e^{2}}{a\beta }}\left(1+{\frac {v_{0}^{2}}{3c^{2}}}\right),\\\\{\frac {dG}{dv_{0}}}&={\frac {2}{3}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}.\\\\{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&={\frac {1}{6}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}\right\}.\end{array}}$.

In this case the integral $\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}$ becomes $\int \sigma _{0}x\left(F_{x}\right)_{0}dS_{0}$, over the surface of the sphere of radius a, and $F_{x}=2\pi \sigma _{0}\cos \theta$.

Thus

${\begin{array}{rl}\int \sigma _{0}x\left(F_{x}\right)_{0}dS_{0}&=4\pi ^{2}a^{3}\sigma _{0}^{2}\int _{0}^{\pi }\cos ^{2}\theta \sin \theta d\theta \\\\&={\frac {8}{3}}\pi a^{3}\sigma _{0}^{2}={\frac {1}{6}}{\frac {e^{2}}{a}},\end{array}}$,

and

${\begin{array}{rl}{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&-{\frac {1}{c^{2}\beta }}\int \sigma _{0}x\left(F_{x}\right)_{0}dS_{0}\\\\&={\frac {1}{6}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}-1+{\frac {v_{0}^{2}}{c^{2}}}\right\}\\\\&={\frac {2}{3}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\\\\&={\frac {dG}{dv_{0}}}.\end{array}}$.