Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/158

VI.

Applying these corrections to the values of $q_{1},q_{2},q_{3}$ compute the second numerical columns under equations ${\text{III}}_{1},{\text{III}}_{2},$ and ${\text{III}}_{3}.$ We do not go on to the computations by equations ${\text{III}}',$ etc., but content ourselves with the old values of $a_{1},b_{1},$ etc., $G,$ and $L,$ which with the new residuals

$\alpha =-.012595,$$\beta =.044949,$$\gamma =.003012,$

give

$C_{1}=-.04567,$$C_{2}=.004952,$$C_{3}=-.04064.$

$\Delta q_{2}=C_{2}-L(Dq_{2}+{\tfrac {1}{2}}C_{2})\Delta q_{2}=.004952-.02322(-.77826+.00247)\Delta q_{2}.$

This gives

$\Delta q_{2}=.005042.$

As the term containing $L$ has increased the value of $\Delta q_{2}$ by .00009, we add this quantity to $C_{1}$ and $C_{3},$ and get

$\Delta q_{1}=-.04558,$$\Delta q_{3}=-.04055.$

With these corrections we compute the third numerical columns under equations ${\text{III}}_{1},$ etc. This time we recompute the quantities $\alpha ',$ etc., with which we repeat the principal computations of IV, and get the new values

$a_{1}=-.01671$	$a_{2}=-.03351815$	$a_{3}=-.0743299$
$b_{1}=+.0149145$	$b_{2}=-.0054413$	$b_{3}=-.0098825$
$c_{1}=-.1576886$	$c_{2}=-.0779570$	$c_{3}=-.0474318$
	$G=.0090929.$

The quantities $H$ and $L$ we neglect as of no consequence at this stage of the approximation.

With these values the new residuals

$\alpha =+.0002919,$$\beta =-.0000044,$$\gamma =+.0000288,$

give

$\Delta q_{1}=C_{1}=+.0010434,$$\Delta q_{2}=C_{2}=+.0013222,$ $\Delta q_{3}=C_{3}=+.0025316.$

These corrections furnish the basis for the fourth columns of numbers under equations ${\text{III}}_{1},$ etc., which give the residuals

$\alpha =+.0000002,$$\beta =+.0000009,$$\gamma =+.0000001,$

and the new corrections

$\Delta q_{1}=+.0000006,$$\Delta q_{2}=+.0000021,$$\Delta q_{3}=+.0000031.$

The corrected values of $q_{1},q_{2},q_{3}$ give

$\log r_{1}=0.428377,$$\log r_{2}=0.4132937,$$\log r_{3}=0.4061399.$

We have carried the approximation farther than is necessary for the following correction of the formula, in order to see exactly where the uncorrected formula would lead us, and for the control afforded by the fourth residuals.