V. 



ON THE DETERMINATION OF ELLIPTIC ORBITS FROM 

 THREE COMPLETE OBSERVATIONS. 



[Memoirs of the National Academy of Sciences, vol. iv. part n. 



pp. 79-104, 1889.] 



THE determination of an orbit from three complete observations 

 by the solution of the equations which represent elliptic motion 

 presents so great difficulties in the general case, that in the first 

 solution of the problem we must generally limit ourselves to the case 

 in which the intervals between the observations are not very long. In 

 this case we substitute some comparatively simple relations between 

 the unknown quantities of the problem, which have an approximate 

 validity for short intervals, for the less manageable relations which 

 rigorously subsist between these quantities. A comparison of the 

 approximate solution thus obtained with the exact laws of elliptic 

 motion will always afford the means of a closer approximation, and 

 by a repetition of this process we may arrive at any required degree 

 of accuracy. 



It is therefore a problem not without interest it is, in fact, the 

 natural point of departure in the study of the determination of orbits 

 to express in a manner combining as far as possible simplicity and 

 accuracy the relations between three positions in an orbit separated 

 by small or moderate intervals. The problem is not entirely deter- 

 minate, for we may lay the greater stress upon simplicity or upon 

 accuracy ; we may seek the most simple relations which are sufficiently 

 accurate to give us any approximation to an orbit, or we may seek 

 the most exact expression of the real relations, which shall not be too 

 complex to be serviceable. 



Derivation of the Fundamental Equation. 



The following very simple considerations afford a vector equation, 

 not very complex and quite amenable to analytical transformation, 

 which expresses the relations between three positions in an orbit 

 separated by small or moderate intervals, with an accuracy far 

 exceeding that of the approximate relations generally used in the 

 determination of orbits. 



