Structural and mechanical assemblies with infinitesimal mobility are studied by means of analytical statics, elaborating and capitalizing on the conceptual difference between virtual and kinematic displacements. The analysis is based on the resolution of a given load into two mutually orthogonal components, which allows the introduction of a statical-kinematic stiffness matrix. The issue of statical-kinematic duality is explored and extended to second-order analysis. It leads to the complementary exclusion condition between kinematic mobility and statical possibility of stable self-stress, and the ensuing statical criterion of immobility. A computationally efficient matrix method for detecting simple first-order mechanisms is presented and compared to an existing method based on the analysis of a certain reduced quadratic form.