ÖFVERSIGT AF K. VBTENSK.-AKAD. FÖRHANDLINGAR 1900, N:0 9. 1071 



(24) 





2 



^ 



a., 



'_) 



«1 1 



^■ 



= 





~ 



+ 









n^ 



ß'i 





/^. ^ 



n 



or 



^(«l/^2 - «2/^1 )' = {ß; + ß'^ ^' + («'i + «D ^' - 2(«,/^, + «,/^2)^^ ■ 



If Ave now eliminate the a and the ß, we get the foilowing 

 equation of the curve 



(25) 



and here is 



V' + 



dH2 

 da? 



C' + h'2 



d-'n\ 



'-'w)^^'} 



dadh 



'§ri 



\Qyi-v- 



V- + 





iß] + .«D 



Taking into consideration the values of the derivatives, we 

 tind that for all centres of libration the curve (25) represents 

 an ellipse. 



In the points L^ , L^ and L^ we have 



dadh 



O 



and one axis of the ellipse tlien coincides with the axis of the 



^-coordinates. I will return to these curves låter on. 



In L^ and L^ the axis of the coordinates may be turned 



in order to coincide with the axis of the ellipse. We therefore 



Substitute 



I = X cos e — y sin ß 



-jj = ^ sin (9 + y cos 6 , 

 and obtain for the ansle e the value 



(26) 



tg2ö = 



2^ 



Dadb 

 3^ W 



With the values of the derivatives in L^ and X.. this 

 equation beconies 



(27) 



tg 2(9 = ± |/3 



ß 



1 + itt' 



where the sign + relätes to L., the sign — ■ to L.. 



