The present paper deals with two problems which arise from the fact that the assumptions involved in the simplest and most useful mathematical solutions of heat conduction are often not obeyed with sufficient accuracy in practice, and some allowance for the discrepancies in therefore necessary. In section (a) is considered the boundary law operating at a lagged surface, and it is shown that when the heat capacity of the lagging is small an approximate boundary law may be used which is essentially similar in form to Newton's law, but contains small additional terms which may, however, be simply taken into account when of sufficient importance. In section (b) is considered the problem of a medium in which the main heat flow is one-dimensional, but there is also subsidiary heat flow in transverse directions, this latter not being, however, of sufficient importance to warrant the use of a cumbersome two- or three-dimensional solution even if one be obtainable. In such cases it is shown that if we concern ourselves only with the temperature averaged in the directions of subsidiary heat flow, then the solution for this latter, correct tho the first order, may be easily obtained as a simple modification of the appropriate one-dimensional solution.