In a nucleosome core particle, the DNA is modelled as an intrinsically straight, isotropic elastic rod which is wrapped around a cylindrical protein kernel of a finite height. A general equilibrium equation is derived for an isotropic inhomogeneous DNA, along which the stiffness varies according to the base sequence. The equation is then solved analytically in terms of the elliptic functions and the elliptic integrals for two fundamental cases: (a) a homogeneous DNA with uniform stiffness along its chain is free of any external forces and moments except the uniform binding force on the surface, (b) a DNA, described in case (a), is subject to a number of concentrated bending moments at different binding sites where the DNA orientation is fixed at a specific direction. Rather strikingly, an equilibrium path found in both cases does not progressively wind around the cylindrical kernel like a helix; instead, it either tends toward the axis of the cylinder or oscillates and intersects itself within a finite region of the cylinder. When the binding energy density is greater than the largest possible elastic energy density, the least energy path can be selected from all the equilibrium solutions under the constraints of the finite height of the protein kernel and no self–intersection of the DNA. As a result, the path with the maximum length of no self–intersection on the finite surface has the lowest energy. In the application to the actual nucleosome core particle, the overall shape and total length of the least energy path on the protein surface calculated for case (a) fail to agree with the experimental observations; whereas, when two binding sites are introduced symmetrically about the middle point of the DNA, i.e. case (b), the theoretical predictions are close to those observed in the crystallographic experiments and the extensive enzyme digestion experiments.