If anyone really wants to consider the case of a 2-conductor confocal
oblate spheroidal capacitor (whose conductors are equal/oppositely
charged) it can be easily be found by superposition using the 1-conductor
solution. Suppose that the two confocal oblate spheroids have a common
focal circle of radius R, that the inner conducting surface has an
eccentricity e_i, and that the outer conducting surface has an
eccentricity e_o, then the capacitance of this capacitor is:
C = 4*[pi]*[epsilon]_0*R/(arcsin(e_i) - arcsin(e_o))