Natural convective flew in a large square enclosure with a narrow rectangular enclosure mounted symmetrically on the upper surface of this main enclosure has been considered. The horizontal bottom and vertical side-walls of the main enclosure are assumed to be at a uniform high temperature and the vertical side-walls of the narrow top enclosure are assumed have a temperature that varies linearly with height above the main enclosure. The top horizontal wall of the upper enclosure is assumed to be adiabatic The situation consider is an approximate model of that which arises when measurements have to be made in a large enclosure containing a hot fluid and in which the measuring instrument is mounted in a small enclosure which is open at one end to the main tank and which has the instrument mounted at the opposite closed end. In order to ensure that the instrument is not exposed to a temperature that is above its operational limit, the walls of this small enclosure are cooled. Fluid properties have been assumed constant except for the density change with temperature that gives rise to the buoyancy forces, this being treated by means of the Boussinesq type approximation. The governing equations have been written in dimensionless form and the resultant dimensionless equations have been solved using a finite-element method. Results have been obtained for a Prandtl number of 0.7. The effects of Rayleigh number and dimensionless height and width of the top enclosure on the maximum temperature of the top adiabatic surface have been investigated. The results show that provided the aspect ratio of the upper enclosure is kept large, overheating of the upper adiabatic surface of the top enclosure is not likely to occur.