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

This gives^[1]

$\Delta q_{1}=-{\frac {({\mathfrak {SS}}''{\mathfrak {S}}''')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}$$\Delta q_{2}=-{\frac {({\mathfrak {SS}}'''{\mathfrak {S}}')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}$$\Delta q_{3}=-{\frac {({\mathfrak {S}}{\mathfrak {S}}'{\mathfrak {S}}'')}{({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')}}\cdot$

(20)

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,

$-{\mathfrak {S}}={\mathfrak {S}}'\Delta q_{1}+{\mathfrak {S}}''\Delta q_{2}+{\mathfrak {S}}'''\Delta q_{3}+{\tfrac {1}{2}}{\mathfrak {T}}'(\Delta q_{1})^{2}+{\tfrac {1}{2}}{\mathfrak {T}}''(\Delta q_{2})^{2}+{\tfrac {1}{2}}{\mathfrak {T}}'''(\Delta q_{3})^{2},$

(21)

where

${\mathfrak {T}}'=$	${\frac {d^{2}{\mathfrak {S}}_{1}}{dq_{1}^{2}}}=$	$2A_{1}B_{1}{\frac {d(r_{1}^{-3})}{dq_{1}}}{\mathfrak {F}}_{1}+$	${\frac {B_{1}}{1+B_{1}r_{1}^{-3}}}{\frac {d^{2}(r_{1}^{-3})}{dq_{1}^{2}}}{\mathfrak {S}}_{1}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (22)
${\mathfrak {T}}''=$	${\frac {d^{2}{\mathfrak {S}}_{2}}{dq_{2}^{2}}}=$	$2B_{2}{\frac {d(r_{2}^{-3})}{dq_{2}}}{\mathfrak {F}}_{2}-$	${\frac {B_{2}}{1-B_{2}r_{2}^{-3}}}{\frac {d^{2}(r_{2}^{-3})}{dq_{2}^{2}}}{\mathfrak {S}}_{2}$
${\mathfrak {T}}'''=$	${\frac {d^{2}{\mathfrak {S}}_{3}}{dq_{3}^{2}}}=$	$2A_{3}B_{3}{\frac {d(r_{3}^{-3})}{dq_{3}}}{\mathfrak {F}}_{3}+$	${\frac {B_{3}}{1+B_{3}r_{3}^{-3}}}{\frac {d^{2}(r_{3}^{-3})}{dq_{3}^{2}}}{\mathfrak {S}}_{3}$

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

$-{\mathfrak {S}}={\mathfrak {S}}'\Delta q_{1}+{\mathfrak {S}}''\Delta q_{2}+{\mathfrak {S}}'''\Delta q_{3}+{\tfrac {1}{2}}{\mathfrak {T}}''\Delta q_{2}^{2}.$

(23)

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

$-({\mathfrak {S}}{\mathfrak {S}}'''{\mathfrak {S}}')=({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')\Delta q_{2}+{\tfrac {6}{10}}({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')\Delta q_{2}^{2}.$

(24)

$({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')$ is easily computed from the formula

$({\mathfrak {T}}''{\mathfrak {S}}'''{\mathfrak {S}}')={\frac {1}{q_{2}}}\left(1-5{\frac {q_{2}^{2}}{r_{2}^{2}}}\right)\left(({\mathfrak {S}}'{\mathfrak {S}}''{\mathfrak {S}}''')+({\mathfrak {F}}_{2}{\mathfrak {S}}'''{\mathfrak {S}}')\right)$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$ (25)
$-{\frac {B_{2}}{q_{2}r_{2}^{3}}}\left(1+{\frac {q_{2}^{2}}{r_{2}^{2}}}\right)({\mathfrak {F}}_{2}{\mathfrak {S}}'''{\mathfrak {S}}'),$

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

1. These equations are obtained by taking the direct products of both members of the preceding equation with ${\mathfrak {S}}''\times {\mathfrak {S}}''',$ ${\mathfrak {S}}'''\times {\mathfrak {S}}',$ and ${\mathfrak {S}}'\times {\mathfrak {S}}'',$ respectively. See footnote on page 119.