An equation for the wavefunction of a three-correlated fermion cluster is obtained from a variational method starting from an explicit expression that takes into account the complete antisymmetrisation between this correlated three-body function and other fermions described by Hartree-Fock orbitals. The Hamiltonian includes both two- and three-body interactions among all nucleons. The condition of the square integrability, which is characteristic of a finite system, is incorporated explicitly. Two sets of equations are derived. The first set is derived under the assumption that nucleons not in the three-body cluster function are described by the Hartree-Fock equations derived by excluding the consideration of the three nucleons forming the cluster. Thus, the effect of the three-body cluster on the Hartree-Fock orbitals of the extra cluster nucleons is neglected. The other set of equations is derived by a complete variation of the total energy with respect to both the three-body correlated cluster function and the wavefunctions of the extra cluster nucleons. Thus, the effect of the three-body cluster on the extra cluster nucleons is taken into account by this set of equations. This latter set of coupled equations is a suitable starting point from which to evaluate the effect of a three-body force on the nuclear binding energy.