The rolling of a railway wheelset along rails without slipping is investigated taking the creep hypothesis into account. The wheelset is represented by two cones that have a common base, and the rails are represented by two circular cylinders with parallel axes. The kinematic characteristics of the unperturbed rolling motion of the wheelset, which occurs when the centre of mass moves along a straight line, and of the perturbed motion, which occurs when the centre of mass of the wheelset describes a sinusoidal trajectory, are determined. The constraint reactions are found for the motions investigated up to small second-order values of the perturbed variables. When the elastic properties of the material in the contact area are taken into account, the creep hypothesis is used, averaging over the fast variables is employed, and the value of the critical speed, above which the rectilinear rolling of the wheelset becomes unstable, is found using averaged equations. In the latter case a periodic mode with two time intervals when the wheel flanges come into contact with the rails is investigated. The reaction force, the work of the dry friction force, and the moment of the active forces needed to maintain the periodic mode are found at the flange/rail contact point within the dry friction model. The boundaries of the stability regions, the parameters of the periodic mode and the moment of the resistance forces as functions of the problem parameters are determined from the formulae obtained by analytical methods.