of the Orbits of Double Stars. 1 8 1 



are not sufficient for the determination of seven elements. 

 Four observations are still more than sufficient. In a similar 

 manner however, to the determination of the orbit of a comet, 

 the superfluous datum of the four observations may be made 

 use of for obtaining, in conjunction with another, a datum which 

 is more suitable to the calculation, or it may be employed for 

 any other particular purpose, and for facilitating the develop- 

 ment. If the tune were not to be taken into consideration at 

 all, it would be necessary, the ellipse having five constants in 

 its most general equation, to have five distances, with their in- 

 termediate angles, in order to obtain the projected ellipse. 

 The condition that the point from which the distances have 

 been taken, is the projection of the focus of the true ellipse, 

 would give the position of the plane, and consequently its 

 true form. If, however, the observations were in other re- 

 spects incomplete ; if, for instance, at a given time only one of 

 the polar coordinates, perhaps only the angle of position, as 

 it is called, had been observed, and as it will be by no means 

 advisable to neglect an element, which is so easily obtained 

 as the time, we should require as many observations as there 

 are elements to be determined; viz. seven. 



The latter case may perhaps arise in future, as, according 

 to our present experience, it is more easy to determine the 

 angle of position with a certain degree of accuracy, than the 

 distance. For the present time, however, it would be super- 

 fluous to found the determination of an orbit on such incom- 

 plete observations, which presents, as it appears, greater diffi- 

 culties than that founded on the employment of complete obser- 

 vations. The epochs are as yet not distant enough from one 

 another to allow the selection of seven of them. I limit my 

 investigations, therefore, to the case of four complete observa- 

 tions, from which the orbit of a double star is to be derived. 



M. Savary has already treated the same subject in the Conn, 

 des Temps for 1830, and has illustrated his methods by an 

 application to Urs. Maj. in such a manner as to present in 

 his calculations a closer conformity to the observations, than 

 could have been expected from the difficulty of obtaining them. 

 The difference of his deduction from that which here follows, 

 consists perhaps only in his employing the properties of the 

 ellipse when referred to its conjugate diameter ; while my in- 

 vestigations are founded on the relations usually employed in 

 astronomy. My formula? may also, perhaps, be more conve- 

 nient for calculation. 



Let the star which is considered to be at rest be the point of 

 beginning of the coordinates, and let the times of the obser- 

 vations in their proper order be t v t^ t& t^ 



The 



