Expressions for the energy balance equation put forward by various authors are reviewed and the need for a more rigorous statement of this equation is pointed out. An equation is deduced which includes rigorous expressions for the conduction and convection losses and an approximate expression for the radiation loss. Another equation which must be solved simultaneously is the hydrodynamical equation of steady flow, which yields the gas drift velocity involved in the convection losses. These general equations, unfortunately, can only be solved for certain special sets of conditions.A solution is given for a special case of a cylindrically symmetrical vertical arc and it is found that on insertion of appropriate numerical values the calculated quantities are in good agreement with experiment.