The theory of second-order elasticity is basically a correction to classical elasticity, but was intended to develop approximate solutions to the complete theory of finite elasticity. The theory was developed by many researchers in the early 1930's, but came into prominence as a result of certain classical experiments and formulations developed by R.S. Rivlin in the 1940's. The theory was applied to examine many elementary states of stress including simple shear and axisymmetric torsion and the importance of the second-order term was clearly established. The application of the theory of second-order elasticity to other problems of engineering interest has been slow primarily because of both mathematical and the algebraic complexity involved in the formulation of problems and their solution. The availability of computer-based Symbolic Mathematical manipulations techniques have made it possible to re-examine a large category of problems that have been ignored in the past. This lecture will present the application of the second-order theory to the torsion of an elastic infinite space induced by the rotation of an embedded spheroidal inclusion. This represents one of the few instances where the second-order theory has been formulated in generalized curvilinear coordinates and applicable to a rubber-like elastic material with a strain energy function of the Mooney-Rivlin form. Through the introduction of a "displacement function", the solution of the second-order problem is reduced to the solution of easily recognizable elliptic partial differential equations of the second- and fourth-order.