Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/152

Components of ${\mathfrak {S}}_{4}$		Components of ${\mathfrak {S}}_{5}$
$\alpha _{4}$	$=\alpha _{3}-\alpha _{1}$	$\alpha _{5}$	$=\alpha _{4}-{\frac {\alpha _{4}\alpha _{2}+\beta _{4}\beta _{2}+\gamma _{4}\gamma _{2}}{s_{2}^{2}}}\alpha _{2}$
$\beta _{4}$	$=\beta _{3}-\beta _{1}$	$\beta _{5}$	$=\beta _{4}-{\frac {\alpha _{4}\alpha _{2}+\beta _{4}\beta _{2}+\gamma _{4}\gamma _{2}}{s_{2}^{2}}}\beta _{2}$
$\gamma _{4}$	$=\gamma _{3}-\gamma _{1}$	$\gamma _{5}$	$=\gamma _{4}-{\frac {\alpha _{4}\alpha _{2}+\beta _{4}\beta _{2}+\gamma _{4}\gamma _{2}}{s_{2}^{2}}}\gamma _{2}$
$s_{4}^{2}$	$=\alpha _{4}^{2}+\beta _{4}^{2}+\gamma _{4}^{2}$	$s_{5}^{2}$	$=\alpha _{5}^{2}+\beta _{5}^{2}+\gamma _{5}^{2}$
$s$	$={\tfrac {1}{2}}(s_{1}+s_{2}+s_{3})$	$S$	$={\tfrac {1}{2}}(s_{1}+s_{4}+s_{3})$

For control only:

$s-s_{2}={\frac {S(S-s_{1})(S-s_{4})(S-s_{3})}{s(s-s_{1})(s-s_{3})}}$

$_{{\text{R}}^{2}}$	$={\frac {S(S-s_{1})(S-s_{4})(S-s_{3})}{s_{2}^{2}}}$	$\tan {\tfrac {1}{2}}(v_{2}-v_{1})$	$={\frac {_{\text{R}}}{s-s_{3}}}$
$N$	$=A_{1}R_{1}+R_{2}+A_{3}R_{3}$	$\tan {\tfrac {1}{2}}(v_{3}-v_{2})$	$={\frac {_{\text{R}}}{s-s_{1}}}$
$p$	$={\frac {2_{{\text{R}}^{2}}s}{N(s-s_{1})(s-s_{3})}}$	$\tan {\tfrac {1}{2}}(v_{3}-v_{1})$	$={\frac {_{\text{R}}s}{(s-s_{1})(s-s_{3})}}$

The computer should be careful to use the corrected values of $A_{1},A_{3}.$ (See VIII.) Trifling errors in the angles should be distributed.

${\begin{aligned}e\sin {\tfrac {1}{2}}(v_{3}+v_{1})&={\frac {{\frac {p}{r_{1}}}-{\frac {p}{r_{3}}}}{2\sin {\tfrac {1}{2}}(v_{3}-v_{1})}}\\e\cos {\tfrac {1}{2}}(v_{3}+v_{1})&={\frac {{\frac {p}{r_{1}}}+{\frac {p}{r_{3}}}-2}{2\cos {\tfrac {1}{2}}(v_{3}-v_{1})}}\\\tan {\tfrac {1}{2}}(v_{3}+v_{1})&e^{2}\end{aligned}}$

$\epsilon ={\sqrt {\frac {1-e}{1+e}}}$$a={\frac {p}{1-e^{2}}}$$b={\sqrt {a/p}}$

Direction-cosines of semi-major axis.
$l$	$=-{\frac {\cos v_{2}}{s_{2}}}\alpha _{2}-{\frac {\sin v_{2}}{s_{5}}}\alpha _{5}$
$m$	$=-{\frac {\cos v_{2}}{s_{2}}}\beta _{2}-{\frac {\sin v_{2}}{s_{5}}}\beta _{5}$
$n$	$=-{\frac {\cos v_{2}}{s_{2}}}\gamma _{2}-{\frac {\sin v_{2}}{s_{5}}}\gamma _{5}$