An analytical solution is presented to the steady-state heat-conduction equation for a heat-generating circular region which is cooled by a ring of equal holes spaced uniformly on a circle concentric with the boundary of the cylinder. The solution is based upon a class of harmonic functions previously defined by Howland. Numerical results are presented in the form of dimensionless curves for the particular examples of (a) uniform heat generation within the region with cooling at the holes and insulated outer boundary, and (b) the shape factor or conductance for this geometry.