ÖFVERSIöT A¥ K. VETENSK.-AKAD. FÖRHANDLING AU 1 !)00, N:0 1>. 1U65 



In all cases we get an algebraic equation of tlie 5th degree 

 tu deterniine q^ or ^.,. These equations are 



jl) ( 1 + ^,)Ql + (3 + 2^/)^* + (3 + f,)Ql — hqI - 2i.,Q., — ^/ - . 



(10*) 2) (\+!ii)Q^^-{6 + 2^i)Q^^ + (S + ^()Ql—^iQl + 21.1 Q., — 1-1=^0. 



\o) (1 + !^i)Qi + (2 + 'd!.i)Ql + (1 + Si-OqI -qI-2q,-1 = . 



Each one of these eqvations has a real positive root; the 

 4 other roots are imaginary. 



Für very small values of /.i these roots have the following 

 limiting values: 



= 7' 



1) ^2 = 1/3 



3) ?i = 1 — Jl ^< • 



H*ence there are generally 5 different points, where small 

 periodic orbits may be generated. These points agree with those 

 points where according to Lagrange it exists an exact Solution 

 of the problem of three bodies. With Gyldén I will call these 

 points centres of libration and design them with L^, Xo, L^, 

 L^ and i^, where L^ and L^ are the points at the vertex of 

 an equilateral triangle on m^m^ and L^, Xj , ig are respectively 

 the points, that are determined by the algebraic equations 1), 

 2), 3) above. 



It is to be observed that periodic orbits may occur also in 

 the vicinity of m^ and m^. 



I will now go to consider the differential equations of the 

 periodic orbits in the vicinity of the centres of libration. 



These equations have the form: 



(11) 



~de- di'' dct^^^ dadh'^' 



W^ ^''di~ dadh^'^ dh'^'^' 



and are thence linear differential equations with constant coeffi- 

 cients. 



