An investigation is made into the state of stress in a closed toroidal shell of uniform thick-walled circular cross section, when loaded by uniform internal and external pressures. The general equations of the classical theory of elasticity, expressed in terms of stress components, are solved approximately by expanding the solutions in power series in a small parameter {R) (a is the external radius of the cross section and R the radius of the axis). This method was used (inter alia) by E. Gohner (1) in his investigation of the twist and pure bending of the sector of a circular ring. The first approximation yields the known solution of the problem of Lame for a thick walled cylinder. The equations for the higher approximations reduce to the problem of plane strain in a circular ring. Only the first three terms of the power series are calculated in this report. The convergence of the series is not investigated.