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

$\Delta q_{3}$						–.80780		–.04055		+.0025316		+ .0000031
$q_{3}$		$+$		3.24945		2.44165		2.40110		2.4036316		2.4036347
$\log r_{3}$		$+$		0.52600		.412217		.4057319		.4061394		.4061399
$\log R_{3}$		$+$		7.98439		8.325742		8.3451948		8.3439733
$\log(1+R_{3})$		$+$		.00417		.009099		.0095108		.0094843
$\log P'''$		$+$		7.91715		8.357016		8.3817516		8.3801993
$\alpha _{3}$		$-$		1.17253		.987590		.9785152		.9790776
$\beta _{3}$		$+$		1.26749		.910305		.8924956		.8936069
$\gamma _{3}$		$-$		.26373		.210171		.2075292		.2076940
$\alpha '''$		$-$		.22847				.2222335
$\beta '''$		$+$		.44441				.4390163
$\gamma '''$		$+$		.06690				.0650888

IV.

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

$a_{1}=-.01254$	$a_{2}=-.03517$	$a_{3}=-.07232$
$b_{1}=+.01726$	$b_{2}=-.00525$	$b_{3}=-.00845$
$c_{1}=-.15746$	$c_{2}=-.08526$	$c_{3}=-.04050$

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

$G=.01006.$

We also get

$H=-.00998,$$L=.02322$[1]

V.

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

$\alpha =-.06058,$$\beta =-.16692,$$\gamma =-.05557.$

From these, with the numbers last computed, we get

$C_{1}=-.65888,$$C_{2}=-.76983,$$C_{3}=-.79939,$

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

$\Delta q_{2}=C_{2}-{\tfrac {6}{10}}L(\Delta q_{2})^{2},$or$\Delta q_{2}=-.76983-.01393(\Delta q_{2})^{2},$

which gives

$\Delta q_{2}=-.77826.$

The quadratic term diminishes the value of $\Delta q_{2}$ by –.00843. Subtracting the same quantity from $C_{1}$ and $C_{2}$ we get

$\Delta q_{1}=-.66731,$$\Delta q_{3}=-.80780.$

1. It would have been better to omit altogether the oaloulation of $H$ and $L,$ if the small value of the latter coidd have been foreseen. In fact, it will be found that the terms containing $L$ hardly improve the convergence, being smaller than quantities which have been neglected. Nevertheless, the use of these terms in this example will illustrate a process which in other cases may be beneficial.