Authors summary: It is well known that though the vanishing of the Wronskian W[ ] of a set of functions on an interval I is a necessary condition for it to be linearly dependent, it is not a sufficient one. Since Peano in 1889 expressed an interest in finding classes of functions for which W[ ] is sufficient for dependence and offered one such example himself, others (M. Bocher, D. R. Curtiss, W. Hurewicz, and G. H. Meisters) have provided related results.
Here the author gives a final answer to the question by first generalizing Peano’s result using the order of a critical point, thereby obtaining a dense set of intervals of dependence. He then shows that W[ ] together with a condition that the intersection of certain subspaces of is nontrivial is equivalent to the linear dependence of on I. The above results are used to establish the dynamical theorem that motion of a particle under the action of a central force field is planar so long as the particle is restricted from the origin. The author provides a counterexample for the case in which the particle passes through the origin.