1 80 Prof. Encke on the Calculation 



and from their combination we obtain the following : 



.-_,) = o, which are the 



</ t~ g 8 



differential equations of an elliptical motion, for which the 

 central force m'+m may be supposed to reside in the one 

 star. The knowledge of the ratio of this mass to that of the 

 sun will not be obtained until the parallax of this pair of stars 

 is known ; nor will the ratio of m and m' be ascertained until 

 we have determined the law, according to which the centres 

 of gravity of the different systems move about, and until by 

 its application we can ascertain the position of that point be- 

 tween the two stars, for which it is true, or of their common 

 centre of gravity*. Our observations do not refer to the el- 

 lipse actually described by the moving star about the one at 

 rest, but to its projection on a plane vertical to the line of 

 vision. The projection of a conic section on a plane any how 

 inclined being likewise a conic section, the projection will in 

 our case likewise be an ellipse, with this difference only, that 

 the star at rest will no longer be in its focus. The determina- 

 tion of the orbit of the double stars is, therefore, on the one hand 

 incomparably more easy than the determination of the orbits 

 of planets, inasmuch as the change of position of the observer 

 does not come into consideration. But, on the other hand, it 

 is more difficult, because the focus itself of the ellipse is 

 not given, but only its projection in the projected ellipse, and 

 the mass being unknown, the measure of areal velocity is 

 wanting. As, therefore, in the case of planets, six data or 

 elements are sufficient, because by them, combined with the 

 known mass of the sun, or the k in Gauss's Theoria Motiis, 

 &c. which depends on it, the area! velocity itself is given, so 

 in the case of double stars this latter must be separately ascer- 

 tained, and we want, consequently, the determination of seven 

 elements. 



Every observation gives, besides the moment of time, two 

 coordinates in the plane of the projected ellipse, which deter- 

 mine the relative position of the moveable stars with regard 

 to the one at rest. Three complete observations or the six 

 quantities obtained by them, and independent of one another, 



* See Bessel, inZach's Monthly Corresp. 1812, August, p. 161. 



are 



