The fundamental solutions required in boundary element analysis‐of torsionai problems of nonhomogeneous transversely isotropic media are derived in this paper. The elastic constants of the nonhomogeneous medium are assumed to be variables as functions of cylindrical coordinates in the formsrαexp (λz) andrα(z+c)β. The displacement resulting from an axisymmetric ring of tangential forces applied in the interior of a infinite space and of a half‐space are obtained. The effects of nonhomogeneity are compared and discussed.