wherefore the major part of the circulation is limited to a region not far removed from the surface of the electron.
The energy of this motion is
½ρ ∫0π ∫a∞ w² · 2π r sin θ · rdθ · dr,
whence, substituting the above value of w, the energy comes out equal to 4/3πρa³w0².
Comparing this with a mass moving with speed u,
m = (8 / 3)πρa³(w0 / u)².
This agrees with the simple hydrodynamic estimate of effective inertia if w0 = ½ √3·u; that is to say, if the whirl in contact with the equator of the sphere is of the same order of magnitude as the velocity of the sphere.
Now for the real relation between w0 and u we must make a hypothesis. If the two are considered equal, the effectively disturbed mass comes out as twice that of the bulk of the electron. If w0 is smaller than u, then the mass of the effectively disturbed fluid is less even than the bulk of an electron; and in that case the estimate of the fluid-density ρ must be exaggerated in order to supply the required energy. It is difficult to suppose the equatorial circulation w0 greater than u, since it is generated by it; and it is most reasonable to treat them both as of the same order of magnitude. So, taking them as equal,
e = a² √(4πρ / μ)
and m = twice the spherical mass.