( 54 ) 



when it i> negative. The curve / = <> Urns consists of hvn branches, 

 both passing through the point ;i = o. 7 1 ' = 2ar, and there exchanging 

 their stability. Since now il lias already been shown that the stable 

 branch T) is. for positive value- of fi, situated on the left side, the 

 unstable branch C must I»' oh the right side. The curve- are 

 represented in fig. 5. 



Our surface has thus been shown to consist 

 of the plane | = and of two sheets, which 

 pass through the line ft = 0, T' = 1st, and 

 then deviate to the left and to the right of 

 the plane T' = 2.-r. The points of the left- 

 hand sheet represent the stable family BE, 

 those of the right-hand sheet the unstable 

 family CC'. This latter sheet therefore inter- 

 sects the plane ;<=<>. J in a curve which on 

 both sides of its point of intersection with 

 the line 5 = bends off towards the right. 

 In this same point of intersection the family 

 A regains its stability, the stable part of the 

 line .: = <>, which represents this family, being enclosed between the 

 two unstable branches of the section just considered. This state of 

 things is rendered in the right-hand pan of fig. 2. Also the form 

 of the section of the surface by a plane T= 7V ~> 2.t. will need 

 no further explanation. It is represented in fig. 47/. "Whether this 

 right hand sheet does reach up to the plane ft = 0.1, so as to pro- 

 duce a real section, cannot be decided by this reasoning. If there 

 is a point of intersection with the line ft = 0.1, § = 0, this must 

 correspond to a value of T' exceeding 414. c 3 = 2.23 jt, since 

 for this value the family .1 is -till unevenly unstable, as is shown 

 by Darwin's work. That the left-hand sheet doe- actually intersect 

 the plane ft = 0.1 is shown by the existence of Darwin- orbit 

 x = — .337. belonging to the family BB' and also by the change 

 of stability of the family .1). 



Thus all results have heen derived which have heen found by 

 Poim'ark in the "Methodes Nouvelles", already quoted. Naturallj 

 Poincakk also must leave the question, whether his results -till hold 

 for ft = 0.1, unanswered. 



It is not uninteresting to consider the solution- />' and C from the 

 point of view of the theory of perturbations. This can. of course, 

 not teach us anything about their stability, but it will give information 

 about the form of the curves / ;/. 7") = <> and iMu. ï) = for small 

 values of >i and §. The period of the undisturbed solution i- 2t. 



