Page:CunninghamPrinciple.djvu/4

Rh 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

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

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

If &xi; &eta; &zeta; 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, &xi;=&beta;x, &eta;=y, &zeta;=z; so that the velocity v1 of the charge relative to the centre is

in the direction of the axis of x.

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

K being the total mechanical force on the electron;

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