It is shown that by choosing a suitable finite difference approximation, parabolic partial differential equations, e.g. the heat conduction equation, can be converted into a series of boundary value problems of the Poisson type, each with specified boundary conditions, which can be easily solved by Southwell's relaxation technique. The method is first discussed with reference to the one-dimensional case, and is then generalized to problems in (x, y)- and (r, z)-co-ordinates. Heat transfer problems can be brought within the scope of the method. The outstanding characteristic of the method is the very stable nature of the solution for all values of the time interval; this permits the choice of relatively great time intervals, so that a complete solution, extending over a long period of time, may be obtained in a small number of steps without loss of accuracy. The required relaxation patterns are given and several numerical examples are included.