Page:CunninghamPrinciple.djvu/4

The assumption on which the theory is built is that the forces from sources exterior to the electron balance those due to the electron itself: this is the assumption that there is no inertia other than electromagnetic, and we deduce the equation

${\frac {dW}{dt}}=-{\frac {dA}{dt}}$,

where W is the electromagnetic energy, and A is the work done by the forces due to the electron itself.

If v₀ is the velocity of the centre of the electron, v=(v₀ + v₁) the velocity of the charge at any point of it, F the mechanical force per unit charge, we have

${\frac {dA}{dt}}=\int \rho (vF)d\tau$,

(vF) being the vector product of v and F.

If ξ η ζ are the coordinates relative to the centre of the electron of the element of charge whose velocity is v, and x y z of the same element when the electron is at rest, ξ=βx, η=y, ζ=z; so that the velocity v₁ of the charge relative to the centre is

${\frac {d\xi }{dt}}=x{\frac {d\beta }{dt}}=-{\frac {xv_{0}}{c^{2}\beta }}{\frac {dv_{0}}{dt}}$

in the direction of the axis of x.

Thus, if F_x is the component of F in that direction,

${\frac {dA}{dt}}=v_{0}\int \rho F_{x}d\tau +\int \rho x{\frac {d\beta }{dt}}F_{x}d\tau$, $=-v_{0}K-{\frac {v_{0}f}{c^{2}\beta }}\int \rho xF_{x}d\tau$,

K being the total mechanical force on the electron;

$=-v_{0}K-{\frac {v_{0}f}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}$;

where the suffix 0 in the last integral refers to the corresponding quantities when the electron is at rest, so that the region of integration is spherical.

For quasistationary motion W is a function of v₀ only, and therefore

${\frac {dW}{dt}}={\frac {dW}{dv_{0}}}{\frac {dv_{0}}{dt}}={\frac {dW}{dv_{0}}}f$.