Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/142

is combined with the correction required by the approximate nature of the equation.^[1]

The solution of the fundamental equation gives us three points, which must necessarily lie in one plane with the sun, and in the lines of sight of the several observations. Through these points we may pass an ellipse, and calculate the intervals of time required by the exact laws of elliptic motion for the passage of the body between them. If these calculated intervals should be identical with the given intervals, corrected for aberration, we would evidently have the true solution of the problem. But suppose, to fix our ideas, that the calculated intervals are a little too long. It is evident that if we repeat our calculations, using in our fundamental equation intervals shortened in the same ratio as the calculated intervals have come out too long, the intervals calculated from the second solution of the fundamental equation must agree almost exactly with the desired values. If necessary, this process may be repeated, and thus any required degree of accuracy may be obtained, whenever the solution of the uncorrected equation gives an approximation to the true positions. For this it is necessary that the intervals should not be too great. It appears, however, from the results of the example of Ceres, given hereafter, in which the heliocentric motion exceeds 62° but the calculated values of the intervals of time differ from the given values by little more than one part in two thousand, that we have here not approached the limit of the application of our formula.

In the usual terminology of the subject, the fundamental equation with intervals uncorrected for aberration represents the first hypothesis; the same equation with the intervals affected by certain numerical coefficients (differing little from unity) represents the second hypothesis; the third hypothesis, should such be necessary, is represented by a similar equation with corrected coefficients, etc.

In the process indicated there are certain economies of labor which should not be left unmentioned, and certain precautions to be observed in order that the neglected figures in our computations may not unduly infiuence the result.

It is evident, in the first place, that for the correction of our fundamental equation we need not trouble ourselves with the position of the orbit in the solar system. The intervals of time, which determine this correction, depend only on the three heliocentric distances $r_{1},r_{2},r_{3}$ and the two heliocentric angles, which will be represented by $v_{2}-v_{1}$ and $v_{3}-v_{2},$ if we write $v_{1},v_{2},v_{3}$ for the true anomalies. These angles ($v_{2}-v_{1}$ and $v_{3}-v_{2}$) may be determined from $r_{1},r_{2},r_{3}$ and $n_{1},n_{2},n_{3},$

1. When an approximate orbit is known in advance, we may correct the fundamental equation at onoe. The formulæ will be given in the Summary, § xii.