Theoretical consideration is given to the equation governing the shapes of molten zones when they are not necessarily surfaces of revolution. This equation is seen to reduce to equations previously given for both vertical and horizontal molten zones when the shape is a surface of revolution. Solution of the equation governing the shape of the vertical molten zone when it is a surface of revolution gives corrected values for the maximum stable zone lengths for large rod radii. Detail consideration is given to the influence of a surface film on vertical zone stability.