1076 CHARLIER, ON PERIODIC ORBITS. 



If we expand also Q in powers of ^ and ij we get, owing 

 to tlie relations (6), 



2n = 2n, + ^^^ + ^ri^ + 2^^^ ^0%, 



where ß^ designs the value of £2 in the centre of libration,, 

 that is in question. 



For the sake of brevity we can choose the point (^^, i^o) 

 in such a manner, that 7]^ is zero, and we have now 



(32) c=2n,+ 





da^ ^nA\dadh\ 



Using the values already given of the derivatives and of 

 the roots we now have in L^ and X^. 



1) (7=2ß, + ,3_|^ 



2) c=2n,-,\-f^. 



We learn from these expressions that values of C some- 

 what greater than 2^3^ belong to the series 1) of curves, or to 

 the family d of curves as I will call these, and for values of C 

 less than 2ß(, we have the family e of curves corresponding to 

 the second root of the equation for v. 



To any value of §q we have two values of C and to any 

 value of C (not difFering much from 212^) there is only one 

 Single value of u". 



As to 2i2„, its value in L^ and L^ is 



(33) 2^0 - 3(1 + itO . 



This is the least value of (7, for which there exist real 

 branches of the curve 



(34) 2ß— C=0, 



which, as M. Hill has first shown, divides the plane in portions, 

 where no motion of a particle can exist, and other portions, 

 where such a motion is possible. 



I will call this curve the limiting curve of motion. 



