1078 



(37) 

 (38) 



CHARLIER, ON PERIODIC ORBITS. 



T 



V6.75/.i 

 T, = r(l + 3.376 (.1) . 



The period of revolution is tlie same tor all curves of the 

 same family. 



For the family e of orbits the period is nearly equal to 7', 

 for the family cl the period is very long. 



Taking for instauce 



iu = 1 : 320000 , 

 so that m^ designs the earth, m^ being the sun. Then we have 



r^ = 217.8 years 



r. = 1 year + 5.546 minutes. 



Returning now to the periodic curves in the vicinity of X, , 

 L.2 and X3, \ve have already found, that their equations are 



, dHl\ ,., / , dHi\ , 



da- 



dir- 



where r- has the same expression as before. The half axes of 

 the ellipses are 



a- = 



52 = 



16nV 



ii'\ + «i 



in consequence of (13*). 



If we here use the expression (29) of ß' + ß'^ through s,>, 

 then these relations are 



(39) ; 





