When analysing the response to treatments administered in sequence to each of a group of subjects it is often advisable to take account of the possibility that the treatments produce residual effects in periods subsequent to their application. Using an appropriate linear model, a family of experimental designs has been found, by which direct treatment effects can be estimated without bias and with minimum variance in exactly the same way as if there were no residual effects. If v treatments are to be tested, the new designs require 2v periods and v2 subjects.