  In Part I of this series, the author has shown how to extend the framework of
Riemannian geometry so as to include infinitesimals of higher than first order.
The purpose of the present contribution is to initiate an investigation into
the implications of higher-order differential geometry for the general theory
of relativity. As we have seen, a novel concept of inertial motion is implied
in the analogue of the geodesic equation when modified to include the effects
of higher infinitesimals and it therefore should not come as a surprise that it
has potentially observable kinematic consequences. The route we prefer to take
goes through the Einstein-Hilbert action generalized to reflect the presence of
higher infinitesimals and a cosmological constant. A variational principle
yields an hierarchy of field equations, which reduce to the Einsteinian case at
first order. In the weak-field limit, we recover the usual relativistic
equation for a moving body to leading order. To exemplify the theoretical
framework, we undertake a preliminary study of the Schwarzschild solution and
Friedmann-Robertson-Walker cosmology in the presence of second-order terms. But
the most exciting results concern the novel effects in orbital mechanics that
arise when the higher-order corrections cannot be neglected. Indeed, the
higher-order Riemannian geometry predicts a modification of Newtonian dynamics
corresponding to the Pioneer anomaly for a spacecraft on a hyperbolic escape
trajectory from the solar system and to the flyby anomaly for the differential
between ingoing and outgoing asymptotic velocities of a spacecraft passing near
a rotating planet -- both of which are found to agree well with sensitive
empirical findings.
