In the paper The existence of unbounded oscillating trajectories in a problem of billiards (1962) Leontovich proved that under bell-like curve (it must be zero at $\pm\infty$) each trajectory oscillates, i.e. it croses y-axis infinitely often. Also he proved that among all trajectories do exist finite and infinite ones.

In the paper The existence of unbounded oscillating trajectories in a problem of billiards (1962) Leontovich proved that under bell-like curve (it must be zero at $\pm\infty$) each trajectory oscillates, i.e. it crosses y-axis infinitely often. Also he proved that among all trajectories do exist finite and infinite ones.