1062 CHARLIER, ON PERIODIC ORBITS. 



To these equations we have tlie Jacobian integral 



where C is a constant and V the velocity of /m relatively to 

 the rotating axes. 



The fiinction i2 is a uniform function of x and y in tlie 

 whole plane with the exception of the points ?nj and m^, the 

 coordinates of which are respectively a; = r^ and x = — r^, the 

 ?/-coordinates being equal to zero. 



Let X = a, y ^= b he the coordinates of an arbitrary point, 

 not coinciding with m^ or m^, it is evident that Q and all its 

 derivatives are susceptible of development in a convergent series 

 in powers of x — a and y — h. If x approaches sufficiently to a 

 and y sufficiently to h, the terms of the lowest dimension in 

 these developraents raust exceed the sum of the remaining terms 

 in the series. 



I will now seek to determine the situation of those points, 

 for which, if we approach sufficiently near to them, periodic Solu- 

 tions of the problem exist, so that the mass m for all time can 

 remain in the vicinity of these points, revolving in curves re- 

 entrant in themselves. 



Let (a, b) be such a point and put 



X =^ a + B, 

 y = b + 71 . 

 It is then 



dt' dt da da^ ' dadb 



(5) 



^ dl_dn .^dHi_ dHl 



I df^ ^ dt'' db '^ ^ dbda ^ ^ db-^ ■*■■•• 



The powers of ^ and r^ higher than the first in the develop- 



ment of -^ and -^ raay be neglected, it we restrict our re- 



search to such periodic orbits, that are situated in the im- 

 raediate vicinity of {a, b). 



