In this paper, a theoretical investigation is made of the steady thermal convection in a two-dimensional fluid layer when it is heated uniformly from below under a simultaneous constraint of non-uniform temperature on its upper surface. Mathematically this is an extention of the method developed in the anthor's previous paper to a problem with inhomogeneous boundary conditions. It was found that the site of spontaneous convection cells is decided according to the surface temperature disturbance having the critical wave length. Surface disturbances having much larger or smaller wave length play very little part in this, while those having wave lengths close to the critical one are effective in determining the general feature of fluid motion.
