  We give an in-depth analysis of a 1-parameter family of electrified droplets
first described in D. Khavinson et. al. (2005). We also investigate a technique
for searching for new solutions to the droplet equation, and rederive via this
technique a 1-parameter family of physical droplets, which were first
discovered by D. Crowdy (1999). We speculate on extensions of these solutions,
in particular to the case of a droplet with multiple connected components.
