A new expression similar to the Huber-von-Mises-Hencky criterion for multiaxial stresses, also known as the distortional energy theory, is derived from the second order stress invariants of the stress tensor, rather than from the deviatoric stress tensor, as in the derivation of the Huber-von Mises-Hencky criterion. It is proposed as a mutually constant design criterion for fatigue resistance under the action of multiaxial cyclic stresses which may be used with either the Gerber-yield criterion or the Goodman-yield (modified Goodman) criterion. As is shown in the following paragraphs, neither of these criteria may be used with the Huber-von Mises-Hencky criterion; it is suited only for the Soderberg criterion which does not involve the ultimate stress.