INTRODUCTION

In my paper on “Periodic Orbits” in the restricted problem of the three bodies (Acta Mathematica, Vol. XXI., 1897, p. 101, or Paper 1) I attempted to make a classification of those orbits which are simply periodic, that is to say, re-entrant after a single circuit either round the Sun, or round the perturbing planet called Jove, or round one of the points of zero force. In that paper I only proposed to consider orbits in which the revolution is in the positive or “direct” sense, and the retrograde orbits were left for future consideration.

Amongst the orbits discussed there was a certain family of satellites which I called A. There were also certain other orbits resembling the figure 8, in which one loop was described round Jove and the second round that point of zero force which lies between the Sun and Jove. The results seemed to indicate that the orbits of these satellites, conjointly with the orbit of a satellite (a) oscillating about the point of zero force, and the figure-of-8 orbit belonged to the same family. But shortly after the publication of the paper it was pointed out both by Mr S. S. Hough and by M. Poincaré that this kind of continuity was not possible, so that the A orbits and the figure-of-8 orbits must in reality belong to different families; and indeed I had independently come to the same conclusion myself.