The equivalence problem for systems of second-order differential equations under point transformations is found to give rise to an {e}-structure of dimension n2 + 4n + 3. It is then shown that the structure function for this {e}-structure is a differential function of two fundamental tensor invariants. The parametric forms of the fundamental invariants are given and their vanishing characterizes the trivial equation ¨xi = 0. We also show that the vanishing of the fundamental invariants characterizes the unique system of second-order ordinary differential equations admitting a maximal-dimension Lie symmetry group. Thus, equations not equivalent to¨xi =0 admit symmetry groups of dimension strictly less than n2 + 4n + 3.