Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/157

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The values of $$\alpha ', \beta ',$$ etc., furnish the basis for the computation of the following quantities:

For we get three values sensibly identical. Adopting the mean, we set We also get

Taking the values of $$\alpha_{1}, \alpha_{2},$$ etc., from the columns under $$\text{III}_{1}, \text{III}_{2}, \text{III}_{3},$$ we form the residuals From these, with the numbers last computed, we get  which might be used as corrections for our values of $$q_{1}, q_{2}, q_{3}.$$ To get more accurate values for these corrections we set  which gives  The quadratic term diminishes the value of $$\Delta q_{2}$$ by –.00843. Subtracting the same quantity from $$C_{1}$$ and $$C_{2}$$ we get