Closed analytic expressions for the vector potential and the magnetic field generated by axisymmetric multipole line currents are presented. These results complement and simplify earlier work utilizing truncated infinite series. The infinite series, which arise by Fourier analyzing the angular dependence of the current distribution, are summed explicitly. For the special case of filamentary line conductors the vector potential, as well as the field components, can be summed. For this case explicit equations for the lines of force and magnetic isobars can be determined. One finds that all lines of force penetrating a cylinder whose radius is determined explicitly by the order of the multipole take on the maximum value of || on that cylinder. Thus, for a low-density plasma composed of adiabatically gyrating charged particles, the region of possible plasma containment within the region bounded by the fixed conductors is a cylinder of circular cross-section with known radius.