Non-physical Solutions in Classical Finite Electron Theory

An approximate linear form of the Peierls-McManus equations of motion for an electron is shown to yield runaway solutions for a particular influence function, which otherwise satisfies all the conditions imposed by the theory. The general problem is to discuss the existence of complex roots of a transcendental equation for an arbitrary form of the influence function. It is shown that this equation always has roots. These, however, may be such that the corresponding motion of the free electron is of a damped nature. It has not been found possible to construct a function which avoids runaway solutions or to prove whether or not such a function exists when all conditions are satisfied.