[ 405 ] 



LXI. On the Calculation of the Orbits of Double Stars. 

 By Professor ENCKE. 



[Continued from page 183.] 



r T*HE first condition to be fulfilled with regard to the four 

 * places, is, that they must all be in an ellipse. If we denote 

 its two semi-axes by a and &, and the excentric anomalies, as 

 they are called, corresponding to the four places, by E, , E 2 , 

 E 3 , E 4 , and assume the axis major of the ellipse as the axis 

 of the abscisses, and the centre of the ellipse as their point of 

 beginning; this condition will be fulfilled, if for the abscisses 

 and ordinates of the four points the following equations take 

 place: 



#[ = a cos E t y l = b sin E t 

 x% = a cos E 2 j/ 2 = b sin E 2 

 ,r 3 = a cos E 3 3/3 =. b sin E 3 

 #4 = a cos E 4 3/4 = b sin E 4 



If we designate the centre of the ellipse by I, we obtain for 

 the double areas of the triangles : 



(I 12) = aft sin E 2 - E t ) 

 (I 13) = a & sin Eg -E,) 

 m (I 1 4) = abs'mE 4 - E,) 



(I 2 3) = a b sin E 3 - E 2 ) 

 (I 24) = a&sinE 4 - E 2 ) 

 (I 34) = abslnE 4 - E 3 ) 



and hence for the double areas of the triangles between the 

 places themselves, 



(1 23) = a{sin(E 2 -E 1 ) + sin (E 3 -E 2 )- sin (E 3 E,)} 

 (1 24) = ab{sm(E^-E l ) + sin(E 4 -E 2 ) sin(E 4 -E J )} 

 (1 34) = ^{sin(E 3 -E 1 ) + sin(E 4 -E 3 )- sin(E 4 -E 1 )} 

 (234) = {sin(E 3 -E 2 ) + sin(E 4 -E 3 )- sin(E 4 -E 2 )} 

 which, by the introduction of this equation, 



E 3 - E, = (E 2 - EJ + (E 3 - E 2 ), 

 and the analogous ones, may be reduced to these forms : 



(1 2 3) = 4 ab sin \ (Ej-Ej) sin 1 (E 3 -E 2 ) sini (E 3 - EJ 

 . . (124) = 4isin 4 (Ea-EJ sin J (E 4 ~E 2 ) sini (E 4 -E,) 

 W (1 3 4) = 4 ab sin (E 3 -E,) sin i (E 4 -E 3 ) sin | (E 4 -E,) 



(2 34)= 4^sini (E 3 -E 2 ) sin \ (E 4 ~E 3 ) sin \ (E 4 -E 2 ) 



The double area of the whole quadrilateral figure will then 

 be found after a small reduction : 



(3) (1 2 3 4) = 4 a b sin \ (E 3 E,) sin J (E 4 -EJ 



For 



