The steady state flow of a viscoelastic second-order fluid in the annular region between two confocal ellipsoids is investigated theoretically. The study is carried out in terms of the three dimensional elliptic (prolate spheroid) coordinates where the inner ellipsoid is rotating with angular velocity about the axis connecting their foci, the z-axis, and the outer ellipsoid is kept at rest . A second-order stream function which describes the planar secondary flow field has been properly formulated by solving the three-dimensional inhomogenious biharmonic vector equation . The particular solution of the planar second-order equation of motion is obtained via the three dimensional Green’s function of the ellipsoidal coordinates . On the basis of the obtained velocity field, the stresses, forces and torques at the boundaries has been calculated for viscoelastic third- order fluids. A third- order resultant torque, which is being as a correction term to a primary viscous one, has been determined .