A multiple integral expansion of the Boltzmann superposition principle when applied to sinusoidal shear oscillations becomes We have evaluated this constitutive equation with hollow cylinders of polycarbonate and polymethylmethacrylate at 1 Hz from Tgto below their 0 transitions. Shear strain amplitude, 70, was increased incrementally from the linear to the nonlinear region. Phase angles and harmonic content were determined with a Rheophasor digital cross correlator. At the maximum strain used, 2-4%, deformation was completely recoverable, after some time, upon returning to the linear region. G1' and G1" vs temperature compare excellently to the literature and our own small strain measurements of G1' and G1" on rectangular bars in free and forced torsion. H3' and H3", and higher harmonic terms were found to be small. All nonlinearity in the range studied can be modeled by G3' and G3" is in the range 10l0-1011 N/m2. G3' is negative and ~ 10 (G3") for both materials. G3' and G3" show a glass transition similar to G1' For polycarbonate they show a very large transition at the Tp of G1'. PMMA showed almost no 0 transition in the nonlinear constants. Molecular explanations and implications for impact and fatigue behavior are discussed as well as potential errors in typical dynamic mechanical data due to these nonlinearities.