1068 



CHAKLIER, ON PERIODIC ORBITS. 



a 



?i 



?2 



/ 



P 



1 



1.6984 



0.6984 



+ 1.570 



+ 2.671 



— 3.531 



0.1 



1.3470 



0.3470 



+ 1.402 



+ 4.695 



— 3.091 



1 : 320000 



1.0101 



0.0101 



+ 1.971 



+ 6.176 



— 4.234 



x. 



a 



Qi 



Qi 



f 



; 



2 



1 



0.6984 



1.6984 



+ 1.570 



+ 2.671 



— 3.531 



0.1 



0.9469 



1.9464 



+ 0.596 



+ 0.253 



— 1.261 



1 : 320000 



1.0000 



2.0000 



+ 0.500 



0.000 



— 1.000 



The limiting values of the roots for evanescent value of (x 

 are given by the expressions: 



r- - 1 ± V28 

 for Xj and Xo, and 



;.2 = _0.5 + 0.5 



for X3. 



As to the points L^ and L^ the discussion is raore easy. 

 With the vahies of the derivatives in these points we get for 

 both the same equation in Z, namely 



(16) 4A* + 4(1 + i.i)r- + 27^t = . 

 The roots of this equation are real, if 



(1 + ^if > 27^1 , 

 that is for 



(17) [.i < 0.0401 . 



If f.1 is smaller than this limiting value, then the values of 

 A- are both negative and there are two different classes of 

 periodic Solutions. If on the contrary f.i > 0.0401, .then no 

 periodic solutions are to be found. 



Mr. BuRRAü has made some numerical researches to find 

 periodic orbits, taking f.i — 1, in the vicinity of L^ and failed 



