Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/139

 This gives  From the corrected values of $$q_{1}, q_{2}, q_{3}$$ we may calculate a new residual $$\mathfrak{S},$$ and from that determine another correction for each of the quantities $$q_{1}, q_{2}, q_{3}.$$ It will sometimes be worth while to use formulæ a little less simple for the sake of a more rapid approximation. Instead of equation (19) we may write, with a higher degree of accuracy, where

It is evident that $$\mathfrak{T}$$ is generally many times greater than $$\mathfrak{T}'$$ or $$\mathfrak{T}',$$ the factor $$B_{2},$$ in the case of equal intervals, being exactly ten times as great as $$A_{1}B_{1}$$ or $$A_{3}B_{3}.$$ This shows, in the first place, that the accurate determination of $$\Delta q_{2}$$ is of the most importance for the subsequent approximations. It also shows that we may attain nearly the same accuracy in writing We may, however, often do a little better than this without using a more complicated equation. For $$\mathfrak{T}' + \mathfrak{T}'$$ may be estimated very roughly as equal to $$\tfrac{1}{2} \mathfrak{T}.$$ Whenever, therefore, $$\Delta q_{1}$$ and $$\Delta q_{3}$$ are about as large as $$\Delta q_{2},$$ as is often the case, it may be a little better to use the coefficient $$\tfrac{6}{10}$$ instead of $$\tfrac{1}{2}$$ in the last term.

For $$\Delta q_{2},$$ then, we have the equation $$(\mathfrak{T} \mathfrak{S}' \mathfrak{S}')$$ is easily computed from the formula

which may be derived from equations (18) and (22).