The non-local kinetics of the electrons in acylindrical, axially symmetric hollow cathode dischargeis theoretically investigated.A new multi-term method for solving the inhomogeneous Boltzmann equationin cylindrical coordinates has been developedto adequately describe the pronounced anisotropyof the electron velocity distribution functionexpected in the cathode fall region.This method is based on a tensorial representation of theexpansion of the electron velocity distribution functionin spherical harmonics.The resultant hierarchy of component equationshas been specified to cylindrical geometrywith rotational symmetry and axial uniformity.A subset of component equations could be separatedwhich determines the relevant distribution partsneeded to analyzethe radial behaviour of important electron propertiesin the hollow cathode.Using measured profiles of the electric field,this subset has numerically been solvedfor a helium plasmaat a discharge current of some mA and a pressure of few torr.Based on this approach in addition to the radial variationof the energy distribution,that of important transport properties and collision ratesis studied and the non-local particle and power balanceof the electrons are analysed.Some theoretical results are compared with available experimental ones.