  We study the evolution of charged droplets of a conducting viscous liquid.
The flow is driven by electrostatic repulsion and capillarity. These droplets
are known to be linearly unstable when the electric charge is above the
Rayleigh critical value. Here we investigate the nonlinear evolution that
develops after the linear regime. Using a boundary elements method, we find
that a perturbed sphere with critical charge evolves into a fusiform shape with
conical tips at time $t_0$, and that the velocity at the tips blows up as
$(t_0-t)^\alpha$, with $\alpha$ close to -1/2. In the neighborhood of the
singularity, the shape of the surface is self-similar, and the asymptotic angle
of the tips is smaller than the opening angle in Taylor cones.
