Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/144

 of the heliocentric distances (say $$r_{2}$$). We need not trouble ourselves farther about this change, for it will be of a magnitude which we neglect in computations with seven-figure tables. That the heliocentric angles thus determined may not agree as closely as they might with the positions on the lines of sight determined by the first solution of the fundamental equation is of no especial consequence in the correction of the fundamental equation, which only requires the exact fulfilment of two conditions, viz:., that our values of the heliocentric distances and angles shall have the relations required by the funda- mental equation to the given intervals of time, and that they shall have the relations required by the exact laws of elliptic motion to the calculated intervals of time. The third condition, that none of these values shall difler too widely from the actual values, is of a looser character.

After the determination of the heliocentric angles and the semi-parameter, the eccentricity and the true anomalies of the three positions may next be determined, and from these the intervals of time. These processes require no especial notice. The appropriate formulæ will be given in the Summary of Formulæ.

Determination of the Orbit from the Three Positions and the Intervals of Time.

The values of the semi-parameter and the heliocentric angles as given in the preceding paragraphs depend upon the quantity $$s_{1} - s_{2} + s_{3},$$ the numerical determination of which from $$s_{1}, s_{2},$$ and $$s_{3},$$ a critical to the second degree when the heliocentric angles are small. This was of no consequence in the process which we have called the correction of the fundamental equation. But for the actual determination of the orbit from the positions given by the corrected equation— or by the uncorrected equation, when we judge that to be sufficient—a more accurate determination of this quantity will generally be necessary. This may be obtained in different ways, of which the following is perhaps the most simple. Let us set and $$s_{4}$$ for the length of the vector $$\mathfrak{S}_{4},$$ obtained by taking the square root of the sum of the squares of the components of the vector. It is evident that $$s_{2}$$ is the longer and $$s_{4}$$ the shorter diagonal of a parallelogram of which the sides are $$s_{1}$$ and $$s_{3}.$$ The area of the triangle having the sides $$s_{1}, s_{2}, s_{3}$$ is therefore equal to that of the triangle having the sides $$s_{1}, s_{3}, s_{4},$$ each being one-half of the parallelogram. This gives