1066 CHARLIER, ON PERIODIC ORBITS. 



A Solution may then be writteu 

 where A and B are deterniined by the equations 



(12) 



.(.--)-.(w.|-) = o 



A Inl 



ll)^4^-P-- 



Fi'om these equations we obtain the following equation for "k 



r- — 



dHi 

 dd- 



%il 



dadh 



%il + 



r — 



dadh 



= 0, 



or 



_.jd^n , d'^Q ,,, , ,\ . d'^Qd^'n ( d'^ny- 



(1^) ^*-^M^" + w-^^' "- ^') + ^' ^- bai)- 



= 0. 



Now the nature of the motion depends on the values of the. 

 roots of this equation. If l- receives a real and negative value, 

 there exists a periodic Solution. If there is no such value, it 

 follows that m can only reniain a finite time in the vicinity of 

 the point (a, h). 



Hence it is necessary to determine the values of the second 

 derivatives of Q for the 5 different centres of libration. 



It is 



da- Qq"^ \ da j dg^ da"^ ßp^ \ da j dq^ da- 



A similar relation exists for 



W 



For the reniaining derivative we have the value 

 d-'Sl _ dHl dQ^ dg^ dQ d^Q, 



dadb Qq^ da dh dg^ dadh 

 d^Qdg.^dg^ dQ d'^g._ 



^p^ da dh dg,, da dh 



