In the final part of his work on Celestial Mechanics which has lately appeared M. Poincaré devotes some space to the consideration of the orbits discussed by Professor Darwin in his recent memoir on Periodic Orbits [Paper 1]. From considerations of analytical continuity M. Poincaré has been driven to the conclusion that Professor Darwin is in error in classifying together certain orbits of the form of a figure-of-8 and others which he designates as satellites of the class A. “Je conclus” says Poincaré “que les satellites A instables ne sont pas la continuation analytique des satellites A stables. Mais alors que sont devenus les satellites A stables?”

Besides the question here raised by Poincaré a second immediately presents itself. After explaining the disappearance of the stable orbits A it is necessary also to give a satisfactory account of the origin of the unstable orbits A. These questions had occupied my mind prior to the publication of M. Poincaré's work, and the present paper contains in substance the conclusions at which I had arrived in connection with them.

It will be seen that the difficulties which have occurred in following up the changes in form of Darwin's orbits arise in some measure from the omission to take into account the orbits described in the present paper as “retrograde,” and the failure to recognize the analytical continuity between these orbits and the direct orbits.