  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.
