The author deals with the mutual inequivalence of g-variation and P-variation of a given action S, the components of linear connection Gamma ^m_kl being required to be symmetric. Under g-variation S is required to be stationary with respect to variations of the metric tensor g_ij, the Gamma ^m_kl being taken to be Christoffel symbols from the outset, whereas under P-variations the g_ij and Gamma ^m_kl are initially regarded as mutually independent of S is required to be stationary with respect to independent variations of these quantities. The discussion is illustrated at length by examples in which the Lagrangian of S is one or another of a set of homogeneous or inhomogeneous quadratic invariants of the Riemann tensor.