When $r$~is diminished the value of~$\gamma$ given by~\Eq{(78.6)} decreases to a minimum for $r = 2a$, and then increases continually becoming infinite at $r = 0$. There is no singularity in the electromagnetic and gravitational fields except at $r = 0$. It is thus possible to have an electron which is strictly a point-singularity, but nevertheless has a finite mass and charge. The solution for the gravitational field of an uncharged particle is quite different in this respect. There is a singularity at $r = 2m$, so that the particle must have a finite perimeter not less than~$4\pi m$. Moreover this singularity is caused by $\gamma$~vanishing, whereas for the point-electron the singularity is due \index{Point-electron}% to $\gamma$~becoming infinite. This demonstration that a point-electron may have exactly the properties which electrons are observed to have is a useful corrective to the general belief that the radius of an electron is known with \emph{certainty}. But on the whole, I think that it is more likely that an electron is a structure of finite size; our solution will then only be valid until we enter the substance of the electron, so that the question of a singularity at the origin does not arise. Assuming that we do not encounter the substance of the electron outside the sphere $r = a$, the total energy of the electromagnetic field beyond this radius would be equal to the mass of the electron determined by observation. \PageSep{187} For this reason $a$~is usually taken as the radius of the electron. If it is admitted that the electromagnetic field continues undisturbed within this limit, an excess of energy accumulates, and it is therefore necessary to suppose that there exists negative energy in the inner portion, or that the effect of the singularity is equivalent to a negative energy. The conception of negative energy is not very welcome according to the usual outlook. Another reason for believing that the charge of an electron is distributed through a volume of radius roughly equal to~$a$ will be found in the investigation of \SecRef{80}. Accordingly I am of opinion that the point-electron is no more than a mathematical curiosity, and that the solution~\Eq{(78.6)} should be limited to values of~$r$ greater than~$a$. \Section{79.}{Electromagnetic action} \index{Action, material or gravitational!electromagnetic}% \index{Electromagnetic action}% \index{Hamiltonian derivative!of electromagnetic action}%