Means are outlined for calculating internal pressure stresses and deformations at or near the junctures of semi-infinite cylindrical vessel shells with hemispherical heads, ellipsoidal heads, toriconical heads, torispherical heads, conical heads, and flat heads. The accurate Love analysis is used in this development. A system of dimensionless variables is proposed for each head shape to define the changes in that shape and the changes in the relative thicknesses of the heads and cylinders. Ways are devised to show the critical head stresses in terms of these dimensionless variables either by graphs or by tabulations. The fundamental theory is sufficiently described to indicate how all results are derived.