Some years ago, it was conjectured that general relativistic shear-free perfect fluids, with a barotropic equation of state such that , must be either expansion-free or irrotational. In this paper we present a fully covariant approach to this conjecture by using the so-called hydrodynamical formalism. We discuss the role of the variables and equations and the procedure for using them in a profitable way. We illustrate the method by giving the proof of the conjecture in two particular cases: (i) when the expansion and the energy density are functionally dependent, i.e. the case studied previously by Lang and Collins using the tetrad formalism; and (ii) when the expansion and the rotation scalar are functionally dependent, a case not treated hitherto.