Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/151

 VIII. For the second hypothesis.

These corrections are to be added to the logarithms of $$A_{1}, A_{3}, B_{1}, B_{2}, B_{3}$$ in equations $$\text{III}_{1}, \text{III}_{2}, \text{III}_{3}$$ and the corrected equations used to correct the values of $$q_{1}, q_{2}, q_{3}$$ until the residuals $$\alpha, \beta, \gamma$$ vanish. The new values of $$A_{1}, A_{3}$$ must satisfy the relation $$A_{1} + A_{3} = 1,$$ and the corrections $$\Delta \log A_{1}, \Delta \log A_{3}$$ must be adjusted, if necessary, for this end.

Third hypothesis. A second correction of equations $$\text{III}_{1}, \text{III}_{2}, \text{III}_{3}$$ may be obtained in the same manner as the first, but this will rarely be necessary.

IX.

Determination of the ellipse. It is supposed that the values of

have been computed by equations $$\text{III}_{1}, \text{III}_{2}, \text{III}_{3}$$ with the greatest exactness, so as to make the residuals $$\alpha, \beta, \gamma$$ vanish, and that the two formulæ for each of the quantities $$s_{1}, s_{2}, s_{3}$$ give sensibly the same value.