Relations for the optimum proportions of multilayer vessels and the computation of stresses are derived. They are based on the energy of distortion-strength theory which is discussed and illustrated by a photographed model. It is shown that the stress distribution in fully or partially plastic cylinders can be obtained as the limit case of a multilayer vessel with an infinite number of layers of constant or variable yield point. Relations are proposed for the determination of the plastic strains. The problem of thermal stresses is discussed and a graphical method for the rapid computation of transient thermal stresses is explained.