182 Prof'.Encke on the Calculation of the Orbits of Double Stars. 



The observations commonly give immediately the p >lar co- 

 ordinates of the moveable star, when the angles are reckoned, 

 sometimes from the circle of declination, sometimes from the 

 parallel of the star at rest. Let the angles reckoned from any 

 one of the principal axes, in the direction of the motion, from 

 to 360, be designated by p l9 /> 2 , p 3 , p 4 ; and let the di- 

 stances be expressed by g } , g 2 g 3 , g 4 . In comparing linear 

 dimensions it is more convenient to have rectangular coordi- 

 nates. If we consider, therefore, the principal axis as the 

 axis of one of the coordinates, and an axis perpendicular to 

 it as that of the others, we have, with due regard to the signs 

 of the trigonometrical functions, 



= ft cospi, & = fa cos p 2 , 3 = p 3 cps#j, 4 = p 4 cos ^ 4 



= ft Sin p l9 >J 2 = p z SU1 P<2 9 *)3 = PS Sm ^35 *)4 = P 



If we designate the origin of the coordinates by 0, and the 

 respective places of the star by 1, 2, 3, 4, and the double areas 

 of the triangles inclosed by any three of these five points, by 

 the respective three numbers in parentheses, we have the fol- 

 lowing six expressions : 



(012) = g, g 2 sin fa-pj = ij 2 f, - >j, 2 

 (0 1 3) = $!& sin (#,-/>,) = % ^ - ^ 3 



/AN (014-) = Sl g 4 sin (Pt-pj = 114?! - >), 4 

 (023) = g a g 3 sin (p 3 - p 9 ) = >j 3 2 - ij 2 ^ 3 



(0 2 4) = 2 g 4 sin (p 4 - 

 (034) = ^ sin 



From their combination the triangles between the places 

 themselves may be derived. We have 



(1 2 3) = (0 1 2) + (0 2 3) - (0 1 3) 

 (1 2 4) = (0 1 2) + (0 2 4) - (0 1 4) 

 (1 3 4) = (0 1 3) +(03 4) -(014) 

 (2 34) = (0 2 3) + (0 3 4) - (0 2 4) 



which, however, are connected together by the following equa- 

 tion of condition : 



(C) (1 2 3 4) = (1 2 3) + (1 3 4) = (1 2 4) + (2 3 4). 



Agreeably to the nature of the ellipse, the signs of the areas 

 (B) must always be positive. A negative sign in the areas 

 (A) denotes that if the triangle be conceived to be formed 

 by the movement of the distance to which the greater index 

 belongs, a movement through an angle of more than 180 in 

 the positive direction has taken place. 



If we denote, in a similar manner, the chords between 



any 



