The geometry of the second order osculating bundle Osc2M, is in many cases determined by its spray and the associated nonlinear connection. For a Banach manifold $M$, we firstly endow $Osc^2M$ with a fiber bundle structure over M. Three different concepts which are used in many infinite dimensional literatures, that is the horizontal distributions, nonlinear connections and sprays are studied in detail and their close interaction is revealed. Moreover we propose a special lift for a connection on the base manifold to $Osc^2$.
