The methodology for the computation of the conductance of a confined arc discharge has been established. A limiting case which describes the capillary argon ion laser at a filling pressure near 0.2 Torr, at a discharge tube radius of 1 mm, and at a discharge current of 7 A has been treated. The theoretical method utilizes a variational principle which resembles the standard approach of Robinson and Bernstein and Nighan. The derivation is carefully explained and applies to the case where the electron mean‐free‐path for momentum transfer is much larger than the radius of the discharge tube. The variational principle formulation has provided a convenient way of deriving computationally simple upper and lower bounds for the electron conductance of the discharge tube. These bounds are particularly important in light of the computation time now required for an accurate calculation.