('53 ) 



We know then that there are stable periodic solutions of the first 

 sort with an arbitrary period, and of the second sort with the period 

 T' = 2 -T and an arbitrary excentricitv. The section therefore consists 

 of the line 5 = and the pari of the line T' = 2it between the 

 points $ = -}-] and | = — J. 1 wish, however, to confine myself 

 to small values of $. This section is represented in Fig. '•'>. 



Next consider the section 



P 



B' 



-J, o 



C' 



III',' 



A4.-0 



Fig. 

 This family 



by a plane T' = 1\' , where 

 1.9* < ?Y< 2.T, and the curves 

 !}>((*, |) = in that plane. The 

 line | = is a part of this 

 curve. The lower part of this 

 line is stable, the upper part is 

 unevenly unstable. In the point 

 where the transition to insta- 

 bility takes place the line§=0 

 is intersected by the branch of 

 i|i = representing the family 

 lai branch must on lioth sides of 



3. 



being liable 



the point of intersection bend upwards, as is represented in fig. 4a. 

 Consider now the section of our surface by a plane parallel to, 

 and at a very small distance from. •? = <*. The orbits represented by 

 the curves /(ft, T') in this plane are all of the second sort. We 

 can imagine these orbits t * ► arise by a variation of << from the un- 

 disturbed periodic orbit of die second sort. They (hen appear as 



A 



c .. 



Fis. la. 



A 

 Fis. lb. 



-A- 



solutions of a problem, in which the parameter is (i, | being kept 

 constant, and thus T (or C) now is our element. These solutions 

 have heen studied by Schwarzschild (Astr. Nachr. 3506). For ft = 

 the period is 2.t. For small values of /< there are (for each value 

 of a) two solutions, viz. B and C when s is positive, B' and C' 



