Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/153



X. Time of perihelion passage.

The threefold determination of $$T$$ affords a control of the exactness of the solution of the problem. If the discrepancies in the values of $$T$$ are such as to require another correction of the formulæ (a third hypothesis), this may be based on the equations where $$T_{(1)}, T_{(2)}, T_{(3)}$$ denote respectively the values obtained from the first, second, and third observations, and $$M$$ the modulus of common logarithms.

XI. For an ephemeris.

These equations are completely controlled by the agreement of the computed and observed positions and the following relations between the constants: