ABSTRACT
This chapter conjectures that any motion of a dynamical system might be approximated by means of those of periodic type, i.e. that the periodic motions would be found to be densely distributed among all possible motions. It becomes a task of the first order of importance for him to determine what the actual distribution of the periodic motions was, so as to prove or disprove his conjecture. For the application of the theorem of Poincare to the periodic motions of more complicated type it is necessary to take account of the fact that every such motion is associated with a distinct second such motion obtained by reversing the direction of motion, although these motions have the same index. In the billiard ball problem, the integral used is that of constant energy, while the auxiliary periodic motions evidently are those obtained by a rolling motion of the billiard ball around the ring in either sense.
