An axisymmetric steady state heat conduction boundary value problem having mixed boundary conditions on both faces of an infinite slab, is reduced to a pair of Fredholm integral equations of the second kind. For large values of h, the slab thickness, a solution correct to O(h -6) i s obtained by expanding the kernels in power series. ยง 1. Introduction. Problems of steady state heat conduction are of great practical importance since in industrial applications it is the steady flow which is of the greatest importance. The solution to the problems of finding the steady temperature in a semi-infinite solid when mixed boundary conditions are prescribed on the circular area p < a and the rest of the face p > a are available in Carslaw and Jaeger's bookl). In this paper the solution is obtained for the problem of a slab when the temperature is prescribed over a circular area p < 1, and there is no flow of heat over p > 1 of one face, and on the other face the flux is prescribed over p < 1, and the rest of the area p > 1 is kept at zero temperature. The mixed boundary conditions give rise to two pairs of dual integral equations which are reduced to two Fredholm integral equations of the second kind. The boundary values of temperature and flux are obtained on the parts of the boundaries, where these are not prescribed. The numerical values of temperature and flux at the boundaries are given in the form of tables and are illustrated graphically. The Fredholm integral equations are solved by the method of successive approximations for a thick slab.
*) Presently at Imperial College, London.