1060 CHARLIER, ON PERIODIC ORBITS. 



seven points are, firstly, tlie two finite masses themselves, in 

 the vicinity of which there exist, as Mr. Darwin bas shown, 

 several series of periodic orbits. The other 5 points agree with 

 the points, where according to Lagrange there exist exact 

 Solutions to the problem of three bodies. 



Of these points 3 lie in a straight line with the finite masses 

 «?j and m^. Through any point in the vicinity of these points 

 a periodic orbit can by drawn. The two remaining points lie 

 on the vertices of the equilateral triangles, that can be described 

 on m^m^. Through any point in their vicinity it is possible, as 

 will be shown in the following, to draw tioo different periodic 

 orbits that satisfy the problem. 



These two Solutions only exist when one of the masses ?7?j 

 and ni^ is sufficiently small in proportion to the other. 



In the other points one has periodic Solutions for all values 

 of the masses. 



The Singular points in straight line with the masses m^ 

 and m, have been the object of research from several authors, 

 Gyldén has pointed out that the periodic orbits in the vicinity 

 of one of these points may be appropriate to explain the Sin- 

 gular phaenomenon known by astronomers under the name of 

 the »Gegenschein». Mr. MouLTON has independently and more 

 fully developed the same idea in an elegant memoir in the 

 Astronomical Journal N:o 483. 



Previous to Mr. Darwin the method of mechanical integra- 

 tion for the research of periodic orbits had been employed by Mr. 

 BuRRAU (Astr. Nachr. 3230 and 3251), who examined the subject 

 when the masses m^ and m., were equal. The periodic curves 

 treated by him belong to the family /> of oscillating satellites. 



For comparison 1 have in the following applied the general 

 formulae to three different assumptions as to the masses, viz. 

 ^==1, jtt = 0.1, jtt = 1 : 320000. The latter being assumed to be 

 the mass of the earth. 



It seems that the method used in this memoir may be ap- 

 plied also to the case, when all three bodies are of finite mass. 



