The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation which is strongly homogeneous in this sense states that the Schwarzian derivative of the dependent variable vanishes. This equation is of impor-tance in the theory of the association between third-order equations and pseudo-Riemannian manifolds due to Newman and his co-workers.