1059 



Öfversigt af Kongl, Vetenskaps-Akadeiiiiens FörluiiKlliiigiir 1!)()(). N:o '.». 



Stockholm. 



Meddol.aiulen frun Lunds Astronomiska Obscrvatorinni. N:o 18. 



On Periodic orbits. 

 By C. V. L. Charlier. 



(Communicated 1900, November 14.) 



We owe to Mr. Hill the introduction of tlie fertile idea 

 of periodic orbits into celestial mechanics. He has himself shown 

 how this theory niay be applied with the greatest advantage to 

 the theory of the raoon, and an elaborated theory on these prin- 

 ciples seems to be prepared for the next time. 



This instance is alone sufficient to show the great practical 

 advantages that may be expected from the theory of periodic 

 orbits. Through the celebrated researches of M. Poincaré this 

 theory has now obtained a rigorous and elegant mathematical 

 foundation. 



Meanwhile, the penetration into this new field of research 

 is by no nieans easy. Though the purely analytical treatment 

 of the problem is now to a great extent possible it seems that 

 most astronomers, who have treated this subject, have preferred 

 to apply the sure but very tardy method of mechanical integra- 

 tion to this problem. Mr. Darwin indeed has succeeded in ob- 

 taining in this manner results of a general character and of the 

 greatest interest. 



Some of his results I will try to obtain in the following in 

 a purely analytical manner. 



It is kuown now, that in the problem of three bodies — 

 one of the bodies being infinitesimal — there exist seven singular 

 points, these points being distinguished by means of the exist- 

 ence of periodic Solutions in their inimediate vicinity. These 



