Different approaches to the evaluation of the momentum flux integral in classical transition state theory (TST) with a generalized reaction coordinate (GRC) have yielded superficially contrasting results for the kinematic Factor which must be included in the ensuing configurational integral. In this work, Smith's recent solution for the momentum flux integral in TST with a GRC (J. Chem. Phys. 1999, 111, 1830) is rederived and generalized in order to establish equivalence with several different approaches in the literature: (a) the pioneering work of Marcus (J. Chem. Phys. 1964, 41, 2624), (b) the reaction path Hamiltonian result of Miller (in Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981), and (c) the flexible TST expression of Robertson, Wagner, and Wardlaw (Faraday Discuss. 1995, 102, 65). We conclude that, while each of these approaches is distinct, the results are consistent. For the general case, a coordinate-dependent kinematic factor does arise from the analytical evaluation of the momentum flux integral, and it is now apparent that this factor can be very simply and efficiently evaluated for incorporation into the ensuing configurational integral on the dividing surface. The same kinematic factor appears in canonical and microcanonical TST expressions for the rate constant, even when total angular momentum is explicitly resolved in the flux integrals.