Conduction of heat between two solids is frequently restricted by surface roughness to flow through microscopic areas of contact. The contact areas can be of any shape, but some idea of their effect can be gained by idealising them as isolated circular areas. Then for conduction between two large, uniform bodies through a contact of radius a there is a 'constriction resistance' due to the concentration of the flow lines equal to 1/2aK where K is the thermal conductivity. This paper investigates how the constriction resistance is modified by a plated coating of different thermal conductivity. The method of finding an upper bound to the resistance by using an arbitrary distribution of heat flux across the contact and calculating and minimising the integral H = integral(v){vertical bar q vertical bar(2)/K}dV (well-known for use in electrical flow problems) is established, and it is shown that this is a highly efficient method of solving the problem, producing answers which while being upper bounds are also highly accurate values, and the method can be strongly recommended for use in other problems. The results for all values of the ratio of spot size to plating thickness aid and of the thermal conductivity ratio kappa equivalent to K-a/K-b are presented in a plot of "universal plating factors". The idea that the heat flux through the contact is simply a combination of the distribution without the coating (q proportional to 1/root a(2) - r(2)) and a uniform flux, although it leads to acceptable values for the constriction resistance, seems not to represent the real physical picture. (C) 2015 Elsevier Ltd. All rights reserved.