The present work is a continuation of the paper by Rosenbluth et al. (Nucl. Fusion 6 (1966) 297) and concerns the investigation of problems associated with the condition for the existence of magnetic surfaces in closed systems of the stellarator type. The unperturbed geometry of the magnetic field is produced by a straight helical field. Exact equations for the motion of the magnetic field lines are put into Hamiltonian form and action-phase variables are introduced. The authors study the effect on magnetic field geometry of a perturbation that is periodic with respect to the co-ordinate along the toroidal axis. It is shown that, if the perturbation is greater than some critical value, the motion of those field lines that are at a sufficient distance from the toroidal axis is similar to Brownian motion and leads to the destruction of the magnetic surfaces. The authors derive the explicit form of the condition forstochastic instability of the magnetic field lines and estimate the coefficient of field line diffusion from the volume. With the above method it is possible to consider the most important form of perturbation in stellarator systems, occurring when the cylindrical, helical field is bent into a torus. Irrespective of the magnitude of the correction for the toroidality of the geometry, destruction of the magnetic surfaces occurs near the separatrix of the unperturbed magnetic field, and this effectively reduces the region bounded by the separatrix of the perturbed field. When the toroidal perturbations are small, the destruction is of a stochastic nature. It follows from the estimate made of the region where destruction takes place that the maximum shear permissible in toroidal systems is substantially limited. In conclusion, the authors present computer calculation results illustrating the stochastic nature of the destruction of the magnetic surfaces.