Laminar free convection from a horizontal cylinder having a prescribed surface heat flux is analysed. The wall surface heat flux is assumed to be varied in the manner of a sub 1 (X/R) square + a sub 2 (X/R) square. Special transformations are devised and employed so that the resulting differential equations and boundary conditions are free of the parameters a sub 1 and a sub 2. These differential equations are solved once and for all; solutions to the original equations for any particular values of a sub 1 and a sub 2 may then be obtained easily as linear combinations of the numerical solutions. It is found that the average Nusselt number for a cylinder having a uniform surface heat flux is larger than that for a cylinder having an isothermal surface by no more than 4 per cent. (Author).