Optimum fully cavitated hydrofoils having analytic profiles, which have either minimum drag for a given lift or maximum lift for a given drag, are investigated. The case of zero cavitation number is considered using the conformal mapping procedure of Levi-Civita, a method ideally suited for the optimization process. The results indicate that profiles with maximum lift for a given drag do not exist, whereas profiles with minimum drag for a given lift exist, but are not unique. The latter optimal profiles have zero drag. It is found further that these profiles have wetted pressures that are not everywhere greater than the cavity pressure or, in this case, greater than the freestream pressure. If the latter condition is imposed a a constraint, it is found that optimum profiles cease to exist. The optimum profiles as found on the other hand, have a significance as ventilated profiles where one has realistic cavitatio pressures (much less than the freestream pressures), but where the cavity pressure lias bee increased artificially to the freestream value by mass addition. Three examples have beei calculated, and the resulting profile shape, the wetted pressure distribution, and the cavity streamlines are given. Lastly, a modification of the Levi-Civita method is indicated for application to nonzero cavitation numbers.