Ambrose, Palais and Singer cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these
results are generalized to a large class of Frechet manifolds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and dissections are equivalent on those Frechet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections.
