Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/134

 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 determinate, 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.

The following very simple considerations afford a vector equation, but 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.