The theory of arcs is developed further by investigating the energy balance equation for high pressure positive columns. The energy balance equation put forward by Suits and based on the Nusselt relation is considered and some refinements in his analysis result in better agreement between theory and experiment. This theory only takes into account energy losses due to convection and conduction, but the neglect of radiation losses is justified for a wide range of conditions by reference to experimental values of radiation efficiencies.The effect of a longitudinal magnetic field on high pressure arcs is then investigated. From the point of view of the energy balance equation the action of a magnetic field is to reduce the contributions of the charged particles to the conduction and convection losses from the column Horizontal and vertical arcs are considered separately, as the nature of the convection, and hence the effect of a magnetic field, is not the same in the two cases.The theory predicts that the action of a magnetic field will tend to a limit with sufficiently strong fields, and certain simple relations are deduced which should apply in this case. According to the theory, when the radius of a high pressure arc is reduced by a magnetic field the arc temperature must rise (incidentally, only by a relatively small amount) and the positive column gradient may increase or decrease, depending on certain relations which are specified There are virtually no experimental data available for these conditions, but the scant data available are in general agreement with the theory