Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/156

$q_{1}$	$=\rho _{1}-.3874081$⁠	$\alpha _{1}$	$=-[8.6700167](q_{1}-9.5901555)(1+R_{1})$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}_{1}$
$r_{1}^{2}$	$=q_{1}^{2}+.8645336$	$\beta _{1}$	$=[9.6833924](q_{1}+.090052)(1+R_{1})$
$R_{1}$	$=[9.6692113]r_{1}^{-3}$	$\gamma _{1}$	$=-[7.9242047](q_{1}+.3874081)(1+R_{1})$
$P'={\frac {[.47712]R_{1}q_{1}}{(1+R_{1})r_{1}^{2}}}$		$\alpha '$	$=-.46775-[8.67002]R_{1}-P'\alpha _{1}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$${\text{III}}'$
		$\beta '$	$=.482383+[9.68339]R_{1}-P'\beta _{1}$
		$\gamma '$	$=-.008399-[7.92420]R_{1}-P'\gamma _{1}$

$\Delta q_{1}$						–.66731		–.04558		–.0010434		+ .0000006
$q_{1}$		$+$		3.22606		2.55875		2.51317		2.5142134		2.5142140
$\log r_{1}$		$+$		.52600		.434960		.4280791		.4282376		.4282377
$\log R_{1}$		$+$		9.09121		8.364331		8.3849740		8.3844985
$\log(1+R_{1})$		$+$		.00533		.009934		.0104122		.0104010
$\log P'$		$+$		8.01967		8.369626		8.3957468		8.3951457
$\alpha _{1}$		$+$		.30136		.336506		.3390605		.3390018
$\beta _{1}$		$+$		1.61938		1.307304		1.286223		1.2867056
$\gamma _{1}$		$-$		.03072		.025316		.0249518		.0249601
$\alpha '$		$-$		.050505		.025316		.0249518		.0249601
$\beta '$		$+$		.47139				.0563438
$\gamma '$		$-$		.00818				.0079821

$q_{2}$	$=\rho _{2}-.9314223$⁠	$\alpha _{2}$	$=+[9.2282738](q_{2}+1.7286820)(1-R_{2})$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}_{2}$
$r_{2}^{2}$	$=q_{2}^{2}+.100681$	$\beta _{2}$	$=-[9.9900800](q_{2}-.0361309)(1-R_{2})$
$R_{2}$	$=[0.38183722]r_{2}^{-3}$	$\gamma _{2}$	$=-[9.1026549](q_{2}-.9314223)(1-R_{2})$
$P''={\frac {[.47712]R_{2}q_{2}}{(1-R_{2})r_{2}^{2}}}$		$\alpha ''$	$=.169151-[9.22827]R_{2}-P''\alpha _{2}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$${\text{III}}''$
		$\beta ''$	$=-.977417+[9.99008]R_{2}-P''\beta _{2}$
		$\gamma ''$	$=-.126664+[9.10265]R_{2}-P''\gamma _{2}$

$\Delta q_{2}$						–.77826		+ .005042		+.0013222		+ .0000021
$q_{2}$		$+$		3.34235		2.56409		2.569132		2.5704542		2.5704563
$\log r_{2}$		$+$		.52600		.412233		.4130733		.4132934		.4132937
$\log R_{2}$		$+$		8.74037		9.081673		9.0791524		9.0784920
$\log(1-R_{2})$		$+$		9.97543		9.944142		9.9444866		9.9445766
$\log P''$		$+$		8.71411		9.199120		9.1954270		9.1944598
$\alpha _{2}$		$+$		.81059		.638489		.6397466		.6400760
$\beta _{2}$		$-$		3.05379		2.172660		2.1787230		2.1803116
$\gamma _{2}$		$-$		.28858		.181843		.1825486		.1827338
$\alpha ''$		$+$		.20182				.2491854
$\beta ''$		$-$		1.08177				1.2018221
$\gamma ''$		$+$		.13464				.1400944

$q_{3}$	$=\rho _{3}-.5599304$⁠	$\alpha _{3}$	$=-[9.3810737](q_{3}+1.5798163)(1+R_{3})$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ ${\text{III}}_{3}$
$r_{3}^{2}$	$=q_{3}^{2}+.7130624$	$\beta _{3}$	$=-[9.6537308](q_{3}-.4630521)(1+R_{3})$
$R_{3}$	$=[9.5623916]r_{3}^{-3}$	$\gamma _{3}$	$=[8.8361256](q_{3}+.5599304)(1+R_{3})$
$P'''={\frac {[.47712]R_{3}q_{3}}{(1-R_{3})r_{3}^{2}}}$		$\alpha '''$	$=-.240477-[9.38107]R_{3}-P'''\alpha _{3}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$${\text{III}}'''$
		$\beta '''$	$=+.450537+[9.65373]R_{3}-P'''\beta _{3}$
		$\gamma '''$	$=+.068569+[8.83613]R_{3}-P'''\gamma _{3}$