Projective connections associated with second-order ODEs

We show that every second-order ODE defines a four-parameter family of projective connections on its two-dimensional solution space. In a special case of ODEs, for which a certain point transformation invariant vanishes, we find that this family of connections always has a preferred representative. This preferred representative turns out to be identical to the projective connection described in Cartanʹs classic paper (Cartan E 1924 Bull. Soc. Math. France 52 205-41, 1955 Oeuvres III 1 825-62).